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Properties

Label 2-608-152.75-c5-0-88
Degree $2$
Conductor $608$
Sign $-0.492 - 0.870i$
Analytic cond. $97.5133$
Root an. cond. $9.87488$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27.1i·3-s + 36.9i·5-s − 150. i·7-s − 496.·9-s + 603.·11-s + 566.·13-s + 1.00e3·15-s − 1.98e3·17-s + (−1.30e3 − 873. i)19-s − 4.09e3·21-s − 1.46e3i·23-s + 1.75e3·25-s + 6.90e3i·27-s + 4.74e3·29-s − 1.90e3·31-s + ⋯
L(s)  = 1  − 1.74i·3-s + 0.661i·5-s − 1.16i·7-s − 2.04·9-s + 1.50·11-s + 0.929·13-s + 1.15·15-s − 1.66·17-s + (−0.831 − 0.555i)19-s − 2.02·21-s − 0.577i·23-s + 0.562·25-s + 1.82i·27-s + 1.04·29-s − 0.356·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(97.5133\)
Root analytic conductor: \(9.87488\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :5/2),\ -0.492 - 0.870i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9766389229\)
\(L(\frac12)\) \(\approx\) \(0.9766389229\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.30e3 + 873. i)T \)
good3 \( 1 + 27.1iT - 243T^{2} \)
5 \( 1 - 36.9iT - 3.12e3T^{2} \)
7 \( 1 + 150. iT - 1.68e4T^{2} \)
11 \( 1 - 603.T + 1.61e5T^{2} \)
13 \( 1 - 566.T + 3.71e5T^{2} \)
17 \( 1 + 1.98e3T + 1.41e6T^{2} \)
23 \( 1 + 1.46e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.74e3T + 2.05e7T^{2} \)
31 \( 1 + 1.90e3T + 2.86e7T^{2} \)
37 \( 1 + 9.12e3T + 6.93e7T^{2} \)
41 \( 1 - 2.64e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.00e4T + 1.47e8T^{2} \)
47 \( 1 + 2.04e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.00e4T + 4.18e8T^{2} \)
59 \( 1 + 1.51e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.82e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.59e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.98e4T + 1.80e9T^{2} \)
73 \( 1 - 2.15e4T + 2.07e9T^{2} \)
79 \( 1 + 7.27e4T + 3.07e9T^{2} \)
83 \( 1 + 1.46e4T + 3.93e9T^{2} \)
89 \( 1 + 1.86e4iT - 5.58e9T^{2} \)
97 \( 1 - 2.09e4iT - 8.58e9T^{2} \)
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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

−8.889097046423806699560180836151, −8.388040448147362832446277553587, −7.09250904747777629362923892054, −6.67900745075433615489823416879, −6.34974303963040765988121195247, −4.49051368630635207988932660503, −3.37639331550380056571231560445, −2.09099706169301444681851606902, −1.19427146884332950061506761668, −0.21545137348545616490746517598, 1.62630396087967892135288577393, 3.09815047112112727471175786352, 4.13870383260370280957915533153, 4.70306667631407017897534513905, 5.78898006124250152516651079103, 6.51226476402711264403625927769, 8.564272806823587517671382567982, 8.824598510401467018481168356300, 9.321727100242648168941655208400, 10.41342204525689585695742119540

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