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Properties

Label 4-63616-1.1-c1e2-0-4
Degree $4$
Conductor $63616$
Sign $-1$
Analytic cond. $4.05621$
Root an. cond. $1.41915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s − 3·17-s + 18-s − 2·23-s − 6·25-s − 28-s − 8·31-s − 32-s + 3·34-s − 36-s − 7·41-s + 2·46-s + 4·47-s − 6·49-s + 6·50-s + 56-s + 8·62-s + 63-s + 64-s − 3·68-s − 71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1/3·9-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.417·23-s − 6/5·25-s − 0.188·28-s − 1.43·31-s − 0.176·32-s + 0.514·34-s − 1/6·36-s − 1.09·41-s + 0.294·46-s + 0.583·47-s − 6/7·49-s + 0.848·50-s + 0.133·56-s + 1.01·62-s + 0.125·63-s + 1/8·64-s − 0.363·68-s − 0.118·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63616\)    =    \(2^{7} \cdot 7 \cdot 71\)
Sign: $-1$
Analytic conductor: \(4.05621\)
Root analytic conductor: \(1.41915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 63616,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
71$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.13.a_ah
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.19.a_ad
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.c_br
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.37.a_ao
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.h_do
43$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.43.a_df
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.ae_cv
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.59.a_ai
61$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.61.a_aba
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.g_bi
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.79.ar_hs
83$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.83.a_adt
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.ae_acw
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−9.575010746501790677345906177134, −9.226791711209052169069620370473, −8.692882584714339402215309294537, −8.163454148013303092898455391897, −7.72607375614043244474530332765, −7.07495631734970734990687090166, −6.66203961635703053324630633155, −5.99137829674169848365705280170, −5.57147900823748360598796861196, −4.79898758502976326318158972719, −3.94579230135018545767189374397, −3.38599014734138675536828510626, −2.45859214469393301212000598768, −1.70191905990015260981552570675, 0, 1.70191905990015260981552570675, 2.45859214469393301212000598768, 3.38599014734138675536828510626, 3.94579230135018545767189374397, 4.79898758502976326318158972719, 5.57147900823748360598796861196, 5.99137829674169848365705280170, 6.66203961635703053324630633155, 7.07495631734970734990687090166, 7.72607375614043244474530332765, 8.163454148013303092898455391897, 8.692882584714339402215309294537, 9.226791711209052169069620370473, 9.575010746501790677345906177134

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