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Properties

Label 2-416-52.35-c2-0-25
Degree $2$
Conductor $416$
Sign $-0.998 - 0.0566i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.02 − 1.17i)3-s − 2.43·5-s + (0.984 − 0.568i)7-s + (−1.75 − 3.04i)9-s + (16.8 + 9.75i)11-s + (7.05 − 10.9i)13-s + (4.94 + 2.85i)15-s + (−11.0 − 19.1i)17-s + (−26.0 + 15.0i)19-s − 2.66·21-s + (−32.0 − 18.4i)23-s − 19.0·25-s + 29.3i·27-s + (−21.5 + 37.3i)29-s + 15.3i·31-s + ⋯
L(s)  = 1  + (−0.676 − 0.390i)3-s − 0.487·5-s + (0.140 − 0.0812i)7-s + (−0.195 − 0.337i)9-s + (1.53 + 0.886i)11-s + (0.542 − 0.839i)13-s + (0.329 + 0.190i)15-s + (−0.649 − 1.12i)17-s + (−1.37 + 0.791i)19-s − 0.126·21-s + (−1.39 − 0.804i)23-s − 0.762·25-s + 1.08i·27-s + (−0.744 + 1.28i)29-s + 0.493i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0566i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.998 - 0.0566i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ -0.998 - 0.0566i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00747923 + 0.263867i\)
\(L(\frac12)\) \(\approx\) \(0.00747923 + 0.263867i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-7.05 + 10.9i)T \)
good3 \( 1 + (2.02 + 1.17i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + 2.43T + 25T^{2} \)
7 \( 1 + (-0.984 + 0.568i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-16.8 - 9.75i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + (11.0 + 19.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (26.0 - 15.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (32.0 + 18.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (21.5 - 37.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 15.3iT - 961T^{2} \)
37 \( 1 + (-25.7 + 44.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (5.36 - 9.29i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8.04 - 4.64i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 27.4iT - 2.20e3T^{2} \)
53 \( 1 + 73.3T + 2.80e3T^{2} \)
59 \( 1 + (18.2 - 10.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.0 + 91.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.45 - 2.57i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (13.0 - 7.51i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 3.70T + 5.32e3T^{2} \)
79 \( 1 + 78.9iT - 6.24e3T^{2} \)
83 \( 1 - 110. iT - 6.88e3T^{2} \)
89 \( 1 + (-17.6 + 30.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (59.2 + 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80247369411397475562040707409, −9.602846889127563378106223910158, −8.713640382889753425476006010740, −7.63289701533753112995928329507, −6.62240244038693935080487456983, −5.99400944744573718786579811396, −4.55982962878544946892254030276, −3.61227972080889638312023897888, −1.71832009655444093832035393094, −0.12175272301501021343719686322, 1.87096716879585161138204348096, 3.89632006855211318516541072334, 4.37073204759588699729468293824, 6.08006942366076996525852072093, 6.31353582852862021308691255722, 7.957736153047829945642637650839, 8.686898147865414148745124792185, 9.670670356131792524875715050102, 10.86278259337773023795695408623, 11.49051324831798185707519512677

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