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Properties

Label 2-304-16.13-c1-0-31
Degree $2$
Conductor $304$
Sign $-0.336 + 0.941i$
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  + (0.987 − 1.01i)2-s + (−0.0665 − 0.0665i)3-s + (−0.0498 − 1.99i)4-s + (−0.733 + 0.733i)5-s + (−0.133 + 0.00165i)6-s − 2.20i·7-s + (−2.07 − 1.92i)8-s − 2.99i·9-s + (0.0182 + 1.46i)10-s + (0.374 − 0.374i)11-s + (−0.129 + 0.136i)12-s + (0.108 + 0.108i)13-s + (−2.22 − 2.17i)14-s + 0.0975·15-s + (−3.99 + 0.199i)16-s + 5.30·17-s + ⋯
L(s)  = 1  + (0.698 − 0.715i)2-s + (−0.0384 − 0.0384i)3-s + (−0.0249 − 0.999i)4-s + (−0.327 + 0.327i)5-s + (−0.0543 + 0.000676i)6-s − 0.832i·7-s + (−0.733 − 0.680i)8-s − 0.997i·9-s + (0.00577 + 0.463i)10-s + (0.112 − 0.112i)11-s + (−0.0374 + 0.0393i)12-s + (0.0301 + 0.0301i)13-s + (−0.595 − 0.581i)14-s + 0.0251·15-s + (−0.998 + 0.0498i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.336 + 0.941i$
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: \((2,\ 304,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961544 - 1.36416i\)
\(L(\frac12)\) \(\approx\) \(0.961544 - 1.36416i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 + 1.01i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.0665 + 0.0665i)T + 3iT^{2} \)
5 \( 1 + (0.733 - 0.733i)T - 5iT^{2} \)
7 \( 1 + 2.20iT - 7T^{2} \)
11 \( 1 + (-0.374 + 0.374i)T - 11iT^{2} \)
13 \( 1 + (-0.108 - 0.108i)T + 13iT^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
23 \( 1 + 2.96iT - 23T^{2} \)
29 \( 1 + (-5.18 - 5.18i)T + 29iT^{2} \)
31 \( 1 + 2.99T + 31T^{2} \)
37 \( 1 + (5.95 - 5.95i)T - 37iT^{2} \)
41 \( 1 - 3.65iT - 41T^{2} \)
43 \( 1 + (-2.17 + 2.17i)T - 43iT^{2} \)
47 \( 1 - 3.00T + 47T^{2} \)
53 \( 1 + (0.619 - 0.619i)T - 53iT^{2} \)
59 \( 1 + (-7.96 + 7.96i)T - 59iT^{2} \)
61 \( 1 + (-7.39 - 7.39i)T + 61iT^{2} \)
67 \( 1 + (-3.76 - 3.76i)T + 67iT^{2} \)
71 \( 1 + 9.60iT - 71T^{2} \)
73 \( 1 + 7.61iT - 73T^{2} \)
79 \( 1 + 16.7T + 79T^{2} \)
83 \( 1 + (-2.91 - 2.91i)T + 83iT^{2} \)
89 \( 1 + 1.61iT - 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−11.59922368150373892098274337822, −10.56714383110845669064436710197, −9.935823346970940063720100386194, −8.796613579823971469970020946445, −7.30710108749618417150771642633, −6.44500621465703550248304269906, −5.22645922553234009600065237337, −3.88943449236512300906130895122, −3.17302803643100830385861746263, −1.10625354267540817656045347925, 2.49834923262296475425554275381, 3.95452352509007092589111361285, 5.16171114739793358166196747417, 5.80684708498947369649520579240, 7.21156411996787861223071128479, 8.060463330191621621024771693852, 8.831347423760101585016127718298, 10.10773392533028966158683079472, 11.44864558641357188686595688904, 12.15794950781008556097749248231

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