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Properties

Label 2-608-152.125-c1-0-9
Degree $2$
Conductor $608$
Sign $0.999 + 0.0157i$
Analytic cond. $4.85490$
Root an. cond. $2.20338$
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  + (0.0638 + 0.0368i)3-s + (1.95 + 1.13i)5-s + 4.70·7-s + (−1.49 − 2.59i)9-s − 1.98i·11-s + (−0.432 + 0.249i)13-s + (0.0833 + 0.144i)15-s + (0.371 − 0.642i)17-s + (1.13 + 4.20i)19-s + (0.300 + 0.173i)21-s + (−1.59 − 2.76i)23-s + (0.0568 + 0.0985i)25-s − 0.441i·27-s + (−5.22 + 3.01i)29-s + 5.44·31-s + ⋯
L(s)  = 1  + (0.0368 + 0.0212i)3-s + (0.875 + 0.505i)5-s + 1.77·7-s + (−0.499 − 0.864i)9-s − 0.597i·11-s + (−0.119 + 0.0692i)13-s + (0.0215 + 0.0372i)15-s + (0.0900 − 0.155i)17-s + (0.261 + 0.965i)19-s + (0.0655 + 0.0378i)21-s + (−0.332 − 0.576i)23-s + (0.0113 + 0.0197i)25-s − 0.0850i·27-s + (−0.970 + 0.560i)29-s + 0.977·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.999 + 0.0157i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.999 + 0.0157i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97964 - 0.0155634i\)
\(L(\frac12)\) \(\approx\) \(1.97964 - 0.0155634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-1.13 - 4.20i)T \)
good3 \( 1 + (-0.0638 - 0.0368i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.95 - 1.13i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.70T + 7T^{2} \)
11 \( 1 + 1.98iT - 11T^{2} \)
13 \( 1 + (0.432 - 0.249i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.642i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.59 + 2.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.22 - 3.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.44T + 31T^{2} \)
37 \( 1 - 10.3iT - 37T^{2} \)
41 \( 1 + (-2.77 + 4.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.56 + 3.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.782 - 1.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.0 + 5.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.10 - 2.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.51 + 0.873i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.42 - 5.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.42 - 9.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.23 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.834 - 1.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 + (2.90 + 5.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.605 - 1.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.56411231445790500641297134581, −9.936037163118143740058876883576, −8.719524735872180758487035424554, −8.213801721162016425239393099751, −7.03878882424877238753747179835, −5.97491435215526241682845814987, −5.29941986456109000242305917282, −4.02291427910883212946596920941, −2.66687551075617953114679783872, −1.42482647055923326336084893263, 1.56157591871142744956513173528, 2.40108165341430700488514239978, 4.37650812885521724126843140344, 5.14989347408941968072324310698, 5.77438339921965059973648339211, 7.35958577443295710526865871475, 7.989584122676605978575369995830, 8.882718864139182557164294723116, 9.718392462746269254803061690157, 10.77384421883740355171400464297

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