Properties

Label 2-304-304.211-c1-0-18
Degree $2$
Conductor $304$
Sign $0.276 + 0.960i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.509 − 1.31i)2-s + (0.00341 + 0.0390i)3-s + (−1.48 + 1.34i)4-s + (−0.335 + 0.718i)5-s + (0.0498 − 0.0244i)6-s + (−0.378 − 0.655i)7-s + (2.52 + 1.26i)8-s + (2.95 − 0.520i)9-s + (1.11 + 0.0759i)10-s + (2.26 − 0.607i)11-s + (−0.0576 − 0.0532i)12-s + (0.294 − 3.36i)13-s + (−0.671 + 0.833i)14-s + (−0.0292 − 0.0106i)15-s + (0.385 − 3.98i)16-s + (1.23 − 6.98i)17-s + ⋯
L(s)  = 1  + (−0.360 − 0.932i)2-s + (0.00197 + 0.0225i)3-s + (−0.740 + 0.672i)4-s + (−0.149 + 0.321i)5-s + (0.0203 − 0.00997i)6-s + (−0.143 − 0.247i)7-s + (0.893 + 0.448i)8-s + (0.984 − 0.173i)9-s + (0.353 + 0.0240i)10-s + (0.684 − 0.183i)11-s + (−0.0166 − 0.0153i)12-s + (0.0816 − 0.933i)13-s + (−0.179 + 0.222i)14-s + (−0.00754 − 0.00274i)15-s + (0.0964 − 0.995i)16-s + (0.298 − 1.69i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.276 + 0.960i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861018 - 0.647888i\)
\(L(\frac12)\) \(\approx\) \(0.861018 - 0.647888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.509 + 1.31i)T \)
19 \( 1 + (-1.52 - 4.08i)T \)
good3 \( 1 + (-0.00341 - 0.0390i)T + (-2.95 + 0.520i)T^{2} \)
5 \( 1 + (0.335 - 0.718i)T + (-3.21 - 3.83i)T^{2} \)
7 \( 1 + (0.378 + 0.655i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.26 + 0.607i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.294 + 3.36i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (-1.23 + 6.98i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.183 + 0.0668i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-4.84 - 3.39i)T + (9.91 + 27.2i)T^{2} \)
31 \( 1 + (2.47 + 4.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.19 + 1.19i)T + 37iT^{2} \)
41 \( 1 + (-1.96 - 1.64i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.24 + 2.91i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (3.65 - 0.644i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.18 - 8.98i)T + (-34.0 + 40.6i)T^{2} \)
59 \( 1 + (-3.75 - 5.36i)T + (-20.1 + 55.4i)T^{2} \)
61 \( 1 + (4.10 + 8.79i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (8.47 - 12.1i)T + (-22.9 - 62.9i)T^{2} \)
71 \( 1 + (-4.20 - 11.5i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-3.34 + 3.98i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (10.7 + 9.00i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-7.93 - 2.12i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (5.77 - 4.84i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (2.07 + 0.365i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.54172889322408003936739181650, −10.48643390954384756946979090995, −9.855060740947192095857043672430, −8.967856314156083894623773698307, −7.72138096227130736200855685150, −6.96276119404731745119728913424, −5.25066282843455318928154793967, −3.95346124732232766669020808616, −2.98461457649276201028379024324, −1.12572859685803780928262452250, 1.50630245607351400565334182272, 4.01579552192755835083056456694, 4.87580786264754965114716477189, 6.31589083228186016183112864951, 6.94304469473605019708674706879, 8.139594126958808858973118825226, 8.945964356168480435605764341166, 9.819062354507016161316485957698, 10.71073039070980526322049348979, 12.06589573109410411109879180772

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