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Properties

Label 4-370e2-1.1-c1e2-0-6
Degree $4$
Conductor $136900$
Sign $1$
Analytic cond. $8.72886$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·5-s − 4·7-s + 4·8-s − 4·10-s + 8·13-s − 8·14-s + 5·16-s − 4·19-s − 6·20-s + 8·23-s − 25-s + 16·26-s − 12·28-s + 14·29-s − 8·31-s + 6·32-s + 8·35-s − 2·37-s − 8·38-s − 8·40-s + 8·43-s + 16·46-s − 4·47-s + 8·49-s − 2·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s + 2.21·13-s − 2.13·14-s + 5/4·16-s − 0.917·19-s − 1.34·20-s + 1.66·23-s − 1/5·25-s + 3.13·26-s − 2.26·28-s + 2.59·29-s − 1.43·31-s + 1.06·32-s + 1.35·35-s − 0.328·37-s − 1.29·38-s − 1.26·40-s + 1.21·43-s + 2.35·46-s − 0.583·47-s + 8/7·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(136900\)    =    \(2^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(8.72886\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 136900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.095884803\)
\(L(\frac12)\) \(\approx\) \(3.095884803\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_i
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_i
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.29.ao_du
31$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bg
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_i
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_c
59$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_i
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ac_c
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2^2$ \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.79.y_lc
83$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_bg
89$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_du
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47718013602246003375742552267, −11.34700688141165288944093319966, −10.91660126747867709884420843737, −10.47355051026309155446020727050, −10.01220533968639049782248084203, −9.281916531279392146854246427040, −8.664288034810394084445726496172, −8.514673371182734911046364075142, −7.72891915311317816813286844699, −7.17433757636585765934880539061, −6.55237808437988363901969639400, −6.41684643507938113959068339258, −5.94187678596989010516857676987, −5.23245744764443935254708726779, −4.56135296400034752490946096296, −4.01037918190529780212636860751, −3.34807128733237594577014040058, −3.34486315504660437022090896820, −2.36705659868046913175965849507, −1.03199153426286123389538234781, 1.03199153426286123389538234781, 2.36705659868046913175965849507, 3.34486315504660437022090896820, 3.34807128733237594577014040058, 4.01037918190529780212636860751, 4.56135296400034752490946096296, 5.23245744764443935254708726779, 5.94187678596989010516857676987, 6.41684643507938113959068339258, 6.55237808437988363901969639400, 7.17433757636585765934880539061, 7.72891915311317816813286844699, 8.514673371182734911046364075142, 8.664288034810394084445726496172, 9.281916531279392146854246427040, 10.01220533968639049782248084203, 10.47355051026309155446020727050, 10.91660126747867709884420843737, 11.34700688141165288944093319966, 11.47718013602246003375742552267

Graph of the $Z$-function along the critical line