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Properties

Label 2-304-16.5-c1-0-12
Degree $2$
Conductor $304$
Sign $0.778 - 0.627i$
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  + (−1.26 + 0.632i)2-s + (1.26 − 1.26i)3-s + (1.19 − 1.60i)4-s + (0.588 + 0.588i)5-s + (−0.797 + 2.39i)6-s + 3.53i·7-s + (−0.502 + 2.78i)8-s − 0.184i·9-s + (−1.11 − 0.371i)10-s + (1.70 + 1.70i)11-s + (−0.507 − 3.53i)12-s + (−2.66 + 2.66i)13-s + (−2.24 − 4.47i)14-s + 1.48·15-s + (−1.12 − 3.83i)16-s + 7.43·17-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)2-s + (0.728 − 0.728i)3-s + (0.599 − 0.800i)4-s + (0.263 + 0.263i)5-s + (−0.325 + 0.977i)6-s + 1.33i·7-s + (−0.177 + 0.984i)8-s − 0.0616i·9-s + (−0.353 − 0.117i)10-s + (0.514 + 0.514i)11-s + (−0.146 − 1.01i)12-s + (−0.737 + 0.737i)13-s + (−0.598 − 1.19i)14-s + 0.383·15-s + (−0.281 − 0.959i)16-s + 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09459 + 0.386014i\)
\(L(\frac12)\) \(\approx\) \(1.09459 + 0.386014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.632i)T \)
19 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.26 + 1.26i)T - 3iT^{2} \)
5 \( 1 + (-0.588 - 0.588i)T + 5iT^{2} \)
7 \( 1 - 3.53iT - 7T^{2} \)
11 \( 1 + (-1.70 - 1.70i)T + 11iT^{2} \)
13 \( 1 + (2.66 - 2.66i)T - 13iT^{2} \)
17 \( 1 - 7.43T + 17T^{2} \)
23 \( 1 + 9.22iT - 23T^{2} \)
29 \( 1 + (0.767 - 0.767i)T - 29iT^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (-4.19 - 4.19i)T + 37iT^{2} \)
41 \( 1 - 2.96iT - 41T^{2} \)
43 \( 1 + (6.52 + 6.52i)T + 43iT^{2} \)
47 \( 1 - 7.69T + 47T^{2} \)
53 \( 1 + (-0.775 - 0.775i)T + 53iT^{2} \)
59 \( 1 + (6.49 + 6.49i)T + 59iT^{2} \)
61 \( 1 + (3.70 - 3.70i)T - 61iT^{2} \)
67 \( 1 + (0.684 - 0.684i)T - 67iT^{2} \)
71 \( 1 + 7.87iT - 71T^{2} \)
73 \( 1 + 5.74iT - 73T^{2} \)
79 \( 1 - 9.25T + 79T^{2} \)
83 \( 1 + (0.337 - 0.337i)T - 83iT^{2} \)
89 \( 1 - 8.02iT - 89T^{2} \)
97 \( 1 + 3.50T + 97T^{2} \)
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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

−12.06041028833235643930375587932, −10.63588260070735040469506764222, −9.616318027904974465916590667536, −8.908742930202440508954060392152, −8.050759745913590991286727563882, −7.16285412701646444755048979997, −6.24954387847722429239956171709, −5.08958537152529411162236670597, −2.70040768989147704978268566816, −1.84913257047641771469482474506, 1.20231539326882648137683229733, 3.29206723130870089115280638925, 3.80402388365556255784322445629, 5.60038304364073756752399600058, 7.34863267693357127056533495413, 7.80635664248285001668207761291, 9.169394025761977055518258115672, 9.671548725664471505159976779579, 10.37156849646054409537312798918, 11.33376230594625192294836996622

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