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Properties

Label 2-608-152.27-c1-0-1
Degree $2$
Conductor $608$
Sign $0.591 - 0.806i$
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  + (−0.705 − 0.407i)3-s + (−3.59 − 2.07i)5-s + 3.24i·7-s + (−1.16 − 2.02i)9-s + 1.21·11-s + (2.01 + 3.48i)13-s + (1.69 + 2.93i)15-s + (1.50 − 2.60i)17-s + (2.16 + 3.78i)19-s + (1.32 − 2.29i)21-s + (−3.01 + 1.73i)23-s + (6.12 + 10.6i)25-s + 4.34i·27-s + (2.35 + 4.08i)29-s + 5.99·31-s + ⋯
L(s)  = 1  + (−0.407 − 0.235i)3-s + (−1.60 − 0.928i)5-s + 1.22i·7-s + (−0.389 − 0.674i)9-s + 0.367·11-s + (0.558 + 0.967i)13-s + (0.436 + 0.756i)15-s + (0.364 − 0.631i)17-s + (0.496 + 0.868i)19-s + (0.288 − 0.499i)21-s + (−0.628 + 0.362i)23-s + (1.22 + 2.12i)25-s + 0.836i·27-s + (0.437 + 0.758i)29-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.591 - 0.806i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657282 + 0.332866i\)
\(L(\frac12)\) \(\approx\) \(0.657282 + 0.332866i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 3.78i)T \)
good3 \( 1 + (0.705 + 0.407i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.59 + 2.07i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.24iT - 7T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + (-2.01 - 3.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.50 + 2.60i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.01 - 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.35 - 4.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.99T + 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 + (-2.97 - 1.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.01 + 1.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.37 - 4.83i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.22 + 2.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.35 - 5.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.300 + 0.173i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.31 - 4.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.581 - 1.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.46 + 2.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.57 - 4.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (12.1 - 7.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.1 - 6.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.30598949638763810632678527023, −9.663845005262580141876308373722, −8.813108682061522681846411008956, −8.350564882723102184036197255012, −7.27572782974857710681598584417, −6.19699585122196755069981886216, −5.27045662859959194955344855136, −4.19615011150767698295510493179, −3.21219152497779520273749613418, −1.20684364919682674072843143692, 0.51525877690075555378786526093, 2.97013297252036172289112597184, 3.88558925221855279211180411862, 4.66914085140982273309818924683, 6.11986661935561984795385610047, 7.06589085522272206516401439569, 7.87852226678293060301271790592, 8.361315043560760277704904733248, 10.17917663441225666346165658601, 10.52799744709462410992589676449

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