Properties

Label 4-260000-1.1-c1e2-0-5
Degree $4$
Conductor $260000$
Sign $1$
Analytic cond. $16.5778$
Root an. cond. $2.01781$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 4·9-s + 3·13-s + 16-s + 6·17-s + 4·18-s + 3·26-s − 6·29-s + 32-s + 6·34-s + 4·36-s − 4·37-s + 14·49-s + 3·52-s − 6·58-s − 2·61-s + 64-s + 6·68-s + 4·72-s − 4·73-s − 4·74-s + 7·81-s + 6·89-s − 16·97-s + 14·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 4/3·9-s + 0.832·13-s + 1/4·16-s + 1.45·17-s + 0.942·18-s + 0.588·26-s − 1.11·29-s + 0.176·32-s + 1.02·34-s + 2/3·36-s − 0.657·37-s + 2·49-s + 0.416·52-s − 0.787·58-s − 0.256·61-s + 1/8·64-s + 0.727·68-s + 0.471·72-s − 0.468·73-s − 0.464·74-s + 7/9·81-s + 0.635·89-s − 1.62·97-s + 1.41·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(260000\)    =    \(2^{5} \cdot 5^{4} \cdot 13\)
Sign: $1$
Analytic conductor: \(16.5778\)
Root analytic conductor: \(2.01781\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 260000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.458615002\)
\(L(\frac12)\) \(\approx\) \(3.458615002\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5 \( 1 \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.43.a_cy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ag_ec
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−9.036215291772226894881286902778, −8.293022393280884028986141782455, −7.895654619618026770158790383149, −7.43565615151402050213631864988, −6.95194397322248815614789721308, −6.65045439006534096204772987427, −5.76689844230494040303576916370, −5.64366171546439695834539309820, −5.03908726049382294565959486319, −4.25412854409389947222151410552, −3.94486851043070574115399103348, −3.43299936688599542205653160876, −2.69639729716473310445901858425, −1.75021024615973518523961957080, −1.15347732004296158210972754165, 1.15347732004296158210972754165, 1.75021024615973518523961957080, 2.69639729716473310445901858425, 3.43299936688599542205653160876, 3.94486851043070574115399103348, 4.25412854409389947222151410552, 5.03908726049382294565959486319, 5.64366171546439695834539309820, 5.76689844230494040303576916370, 6.65045439006534096204772987427, 6.95194397322248815614789721308, 7.43565615151402050213631864988, 7.895654619618026770158790383149, 8.293022393280884028986141782455, 9.036215291772226894881286902778

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