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Properties

Label 2-416-13.9-c3-0-11
Degree $2$
Conductor $416$
Sign $0.522 - 0.852i$
Analytic cond. $24.5447$
Root an. cond. $4.95427$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.25 + 7.37i)3-s − 9.83·5-s + (−12.0 − 20.8i)7-s + (−22.7 − 39.4i)9-s + (14.2 − 24.5i)11-s + (−13 + 45.0i)13-s + (41.8 − 72.4i)15-s + (16.5 + 28.7i)17-s + (3.54 + 6.13i)19-s + 205.·21-s + (98.6 − 170. i)23-s − 28.3·25-s + 157.·27-s + (−11.3 + 19.6i)29-s − 48.3·31-s + ⋯
L(s)  = 1  + (−0.819 + 1.41i)3-s − 0.879·5-s + (−0.651 − 1.12i)7-s + (−0.842 − 1.45i)9-s + (0.389 − 0.674i)11-s + (−0.277 + 0.960i)13-s + (0.720 − 1.24i)15-s + (0.236 + 0.409i)17-s + (0.0427 + 0.0740i)19-s + 2.13·21-s + (0.894 − 1.54i)23-s − 0.226·25-s + 1.12·27-s + (−0.0725 + 0.125i)29-s − 0.280·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(24.5447\)
Root analytic conductor: \(4.95427\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :3/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7785938802\)
\(L(\frac12)\) \(\approx\) \(0.7785938802\)
\(L(\frac{5}{2})\) 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 \( 1 + (13 - 45.0i)T \)
good3 \( 1 + (4.25 - 7.37i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 9.83T + 125T^{2} \)
7 \( 1 + (12.0 + 20.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-14.2 + 24.5i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-16.5 - 28.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.54 - 6.13i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-98.6 + 170. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (11.3 - 19.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 48.3T + 2.97e4T^{2} \)
37 \( 1 + (7.58 - 13.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (237. - 411. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-120. - 208. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 214.T + 1.03e5T^{2} \)
53 \( 1 - 719.T + 1.48e5T^{2} \)
59 \( 1 + (-71.6 - 124. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (258. + 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (308. - 534. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (131. + 228. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 1.05e3T + 3.89e5T^{2} \)
79 \( 1 - 1.25e3T + 4.93e5T^{2} \)
83 \( 1 + 224.T + 5.71e5T^{2} \)
89 \( 1 + (635. - 1.10e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (398. + 689. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.94147054069170432788316955122, −10.16436523937633626800816881218, −9.394569378592593233637952054818, −8.338897299498019244999422892129, −6.99962333911680142738080591749, −6.19965827432163144349277213051, −4.80176837604238711604571138242, −4.08804058762572754142172369288, −3.37144041456580217387998442741, −0.61802569025816099976861180147, 0.54587146524662132648326596620, 2.06548669051693363237331724743, 3.41176518770997373352972770374, 5.22096385699861503563605393229, 5.88124959958777562870168700374, 7.10410522347820295739423686204, 7.48776463648528268705690500215, 8.641527818305051535226346715958, 9.707210359031379128481366337696, 11.01221484907962099997463386450

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