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Properties

Label 2-304-16.13-c1-0-23
Degree $2$
Conductor $304$
Sign $0.926 + 0.377i$
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  + (−1.30 + 0.543i)2-s + (0.641 + 0.641i)3-s + (1.41 − 1.41i)4-s + (2.46 − 2.46i)5-s + (−1.18 − 0.489i)6-s + 0.804i·7-s + (−1.07 + 2.61i)8-s − 2.17i·9-s + (−1.87 + 4.55i)10-s + (2.23 − 2.23i)11-s + (1.81 − 0.00537i)12-s + (−3.45 − 3.45i)13-s + (−0.437 − 1.05i)14-s + 3.15·15-s + (−0.0237 − 3.99i)16-s − 3.81·17-s + ⋯
L(s)  = 1  + (−0.923 + 0.384i)2-s + (0.370 + 0.370i)3-s + (0.705 − 0.709i)4-s + (1.10 − 1.10i)5-s + (−0.483 − 0.199i)6-s + 0.304i·7-s + (−0.378 + 0.925i)8-s − 0.725i·9-s + (−0.593 + 1.43i)10-s + (0.675 − 0.675i)11-s + (0.523 − 0.00155i)12-s + (−0.957 − 0.957i)13-s + (−0.116 − 0.280i)14-s + 0.815·15-s + (−0.00592 − 0.999i)16-s − 0.925·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.926 + 0.377i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10752 - 0.216889i\)
\(L(\frac12)\) \(\approx\) \(1.10752 - 0.216889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.543i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.641 - 0.641i)T + 3iT^{2} \)
5 \( 1 + (-2.46 + 2.46i)T - 5iT^{2} \)
7 \( 1 - 0.804iT - 7T^{2} \)
11 \( 1 + (-2.23 + 2.23i)T - 11iT^{2} \)
13 \( 1 + (3.45 + 3.45i)T + 13iT^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 + (-6.51 - 6.51i)T + 29iT^{2} \)
31 \( 1 - 7.13T + 31T^{2} \)
37 \( 1 + (1.33 - 1.33i)T - 37iT^{2} \)
41 \( 1 - 5.62iT - 41T^{2} \)
43 \( 1 + (0.204 - 0.204i)T - 43iT^{2} \)
47 \( 1 + 6.17T + 47T^{2} \)
53 \( 1 + (3.93 - 3.93i)T - 53iT^{2} \)
59 \( 1 + (-0.172 + 0.172i)T - 59iT^{2} \)
61 \( 1 + (-3.14 - 3.14i)T + 61iT^{2} \)
67 \( 1 + (-4.31 - 4.31i)T + 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \)
89 \( 1 + 15.4iT - 89T^{2} \)
97 \( 1 + 9.57T + 97T^{2} \)
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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

−11.58357145665013897447777220206, −10.23172460580681760655624563769, −9.580448246346112576111266936574, −8.903632991140097198306963230710, −8.297452692647032545663612471456, −6.74227207220361008400787417160, −5.82332749173176755748270248693, −4.84262361552867902187579390428, −2.82289414666537164380954879970, −1.16910576933223548415467185646, 2.02754299072603276716411462632, 2.55783507589906388740892210991, 4.44640855454669939622051635137, 6.61440123518062703277045975866, 6.77564918996203911757927287795, 8.005005593995415501476736852931, 9.118336995130171979888891108964, 10.02809550083442884865369021150, 10.52237732062688953363442767709, 11.58600349594620164924881636270

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