Properties

Label 4-392e2-1.1-c1e2-0-4
Degree $4$
Conductor $153664$
Sign $1$
Analytic cond. $9.79774$
Root an. cond. $1.76921$
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  + 2·9-s − 8·11-s − 2·25-s + 4·29-s + 20·37-s − 8·43-s + 12·53-s + 24·67-s + 16·79-s − 5·81-s − 16·99-s − 24·107-s + 4·109-s + 20·113-s + 26·121-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.41·11-s − 2/5·25-s + 0.742·29-s + 3.28·37-s − 1.21·43-s + 1.64·53-s + 2.93·67-s + 1.80·79-s − 5/9·81-s − 1.60·99-s − 2.32·107-s + 0.383·109-s + 1.88·113-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(153664\)    =    \(2^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(9.79774\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 153664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.371716252\)
\(L(\frac12)\) \(\approx\) \(1.371716252\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.47.a_ck
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.59.a_eg
61$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.61.a_ada
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 + 162 T^{2} + p^{2} T^{4} \) 2.97.a_gg
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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

−11.45557698899408401702952962480, −11.02003112888352876795354260785, −10.62215714150156560844700483300, −10.06505285963591712916076259719, −9.883554933290803053469737725338, −9.445249646624673432647354616810, −8.648320943661838632975628604705, −8.186496361801529615121767807243, −7.77258515529897134687955321671, −7.59171894925612765638676788058, −6.75192047741729539144599946478, −6.43003990518997189359468810491, −5.45943299544792540458308659981, −5.44719222554173891922828470993, −4.62505901036478445986136556659, −4.21172775218603880989909672935, −3.34243604037230066278060279781, −2.58900169588139173158210362787, −2.19527274733034608976408563982, −0.78239876731257050278730621113, 0.78239876731257050278730621113, 2.19527274733034608976408563982, 2.58900169588139173158210362787, 3.34243604037230066278060279781, 4.21172775218603880989909672935, 4.62505901036478445986136556659, 5.44719222554173891922828470993, 5.45943299544792540458308659981, 6.43003990518997189359468810491, 6.75192047741729539144599946478, 7.59171894925612765638676788058, 7.77258515529897134687955321671, 8.186496361801529615121767807243, 8.648320943661838632975628604705, 9.445249646624673432647354616810, 9.883554933290803053469737725338, 10.06505285963591712916076259719, 10.62215714150156560844700483300, 11.02003112888352876795354260785, 11.45557698899408401702952962480

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