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Properties

Label 2-304-16.13-c1-0-1
Degree $2$
Conductor $304$
Sign $0.982 - 0.187i$
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  + (−0.724 − 1.21i)2-s + (−2.33 − 2.33i)3-s + (−0.949 + 1.76i)4-s + (0.751 − 0.751i)5-s + (−1.14 + 4.53i)6-s + 3.10i·7-s + (2.82 − 0.122i)8-s + 7.92i·9-s + (−1.45 − 0.368i)10-s + (−2.88 + 2.88i)11-s + (6.33 − 1.89i)12-s + (−1.54 − 1.54i)13-s + (3.77 − 2.25i)14-s − 3.51·15-s + (−2.19 − 3.34i)16-s + 3.53·17-s + ⋯
L(s)  = 1  + (−0.512 − 0.858i)2-s + (−1.34 − 1.34i)3-s + (−0.474 + 0.880i)4-s + (0.336 − 0.336i)5-s + (−0.467 + 1.85i)6-s + 1.17i·7-s + (0.999 − 0.0433i)8-s + 2.64i·9-s + (−0.461 − 0.116i)10-s + (−0.869 + 0.869i)11-s + (1.82 − 0.547i)12-s + (−0.427 − 0.427i)13-s + (1.00 − 0.601i)14-s − 0.907·15-s + (−0.549 − 0.835i)16-s + 0.857·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.982 - 0.187i$
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: \((2,\ 304,\ (\ :1/2),\ 0.982 - 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378952 + 0.0358634i\)
\(L(\frac12)\) \(\approx\) \(0.378952 + 0.0358634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.724 + 1.21i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (2.33 + 2.33i)T + 3iT^{2} \)
5 \( 1 + (-0.751 + 0.751i)T - 5iT^{2} \)
7 \( 1 - 3.10iT - 7T^{2} \)
11 \( 1 + (2.88 - 2.88i)T - 11iT^{2} \)
13 \( 1 + (1.54 + 1.54i)T + 13iT^{2} \)
17 \( 1 - 3.53T + 17T^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
29 \( 1 + (-4.10 - 4.10i)T + 29iT^{2} \)
31 \( 1 + 8.86T + 31T^{2} \)
37 \( 1 + (5.87 - 5.87i)T - 37iT^{2} \)
41 \( 1 + 1.90iT - 41T^{2} \)
43 \( 1 + (-1.59 + 1.59i)T - 43iT^{2} \)
47 \( 1 - 9.75T + 47T^{2} \)
53 \( 1 + (4.03 - 4.03i)T - 53iT^{2} \)
59 \( 1 + (7.96 - 7.96i)T - 59iT^{2} \)
61 \( 1 + (2.91 + 2.91i)T + 61iT^{2} \)
67 \( 1 + (-7.69 - 7.69i)T + 67iT^{2} \)
71 \( 1 - 7.79iT - 71T^{2} \)
73 \( 1 + 3.45iT - 73T^{2} \)
79 \( 1 + 0.238T + 79T^{2} \)
83 \( 1 + (-6.68 - 6.68i)T + 83iT^{2} \)
89 \( 1 + 0.325iT - 89T^{2} \)
97 \( 1 + 9.17T + 97T^{2} \)
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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

−12.00325190874043249293397585514, −10.94783602074138175186747605472, −10.17316653828883723837794152551, −8.960431975751196956389095026547, −7.80990744489252518529724286073, −7.10529342225056890617717156970, −5.55665523350332250548651284668, −5.08639023503350074968099012887, −2.57783154535286735552438969150, −1.50325409218188584037416731360, 0.40436506226383474764134559117, 3.81213803040901964678484342592, 4.87466336927626937547616808882, 5.75660423867300569589325231038, 6.57611945173323588707534101196, 7.71009662279197562175683391778, 9.126398600702232739514827455780, 10.06630185629448258216462025274, 10.54749796693590551749957911518, 11.08995126689734953194949043078

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