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Properties

Label 2-308-77.62-c1-0-5
Degree $2$
Conductor $308$
Sign $0.998 - 0.0448i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (1.51 + 0.493i)3-s + (0.260 + 0.359i)5-s + (1.55 − 2.14i)7-s + (−0.366 − 0.266i)9-s + (2.44 − 2.23i)11-s + (2.91 + 2.12i)13-s + (0.218 + 0.673i)15-s + (−1.73 + 1.26i)17-s + (−2.60 + 8.03i)19-s + (3.41 − 2.48i)21-s + 1.73·23-s + (1.48 − 4.56i)25-s + (−3.23 − 4.45i)27-s + (−6.19 + 2.01i)29-s + (−2.85 + 3.92i)31-s + ⋯
L(s)  = 1  + (0.876 + 0.284i)3-s + (0.116 + 0.160i)5-s + (0.586 − 0.809i)7-s + (−0.122 − 0.0887i)9-s + (0.737 − 0.675i)11-s + (0.809 + 0.588i)13-s + (0.0565 + 0.174i)15-s + (−0.420 + 0.305i)17-s + (−0.598 + 1.84i)19-s + (0.744 − 0.542i)21-s + 0.361·23-s + (0.296 − 0.913i)25-s + (−0.623 − 0.857i)27-s + (−1.14 + 0.373i)29-s + (−0.512 + 0.704i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.998 - 0.0448i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.998 - 0.0448i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80757 + 0.0405194i\)
\(L(\frac12)\) \(\approx\) \(1.80757 + 0.0405194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.55 + 2.14i)T \)
11 \( 1 + (-2.44 + 2.23i)T \)
good3 \( 1 + (-1.51 - 0.493i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.260 - 0.359i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-2.91 - 2.12i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.73 - 1.26i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.60 - 8.03i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + (6.19 - 2.01i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.85 - 3.92i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.305 - 0.940i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.71 + 8.34i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.32iT - 43T^{2} \)
47 \( 1 + (6.63 + 2.15i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.09 + 5.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-10.2 + 3.34i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.62 - 4.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + (10.6 - 7.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.23 + 6.86i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.15 - 7.09i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.6 - 9.18i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + (-0.726 + 0.999i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.48946665675207452303822105067, −10.79710365219596578119037096289, −9.775094807364677980196812668849, −8.667883132698679208124747923463, −8.239592491527137936577713275462, −6.87359367440994241056191017641, −5.85614600497781662585524657917, −4.13296654115164359096352753969, −3.51595144571786382892682246850, −1.70881354820295562751692956296, 1.84394543557431530139024722311, 2.99066430975550622045664654459, 4.54872637537364151785604473613, 5.69958359721651625878759432234, 7.00643641130979328005223527821, 8.012848806444426398356952700913, 8.981123772692025053811551622293, 9.316993285190711213719986278807, 11.05952192879820019891286384545, 11.49102211392283986784796800991

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