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Properties

Label 2-608-19.18-c2-0-29
Degree $2$
Conductor $608$
Sign $-0.248 + 0.968i$
Analytic cond. $16.5668$
Root an. cond. $4.07023$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48i·3-s − 4.43·5-s + 2.39·7-s + 6.80·9-s − 12.1·11-s − 3.28i·13-s − 6.57i·15-s − 7.93·17-s + (−18.4 − 4.71i)19-s + 3.54i·21-s + 22.6·23-s − 5.33·25-s + 23.4i·27-s − 42.2i·29-s + 6.86i·31-s + ⋯
L(s)  = 1  + 0.494i·3-s − 0.887·5-s + 0.341·7-s + 0.755·9-s − 1.10·11-s − 0.252i·13-s − 0.438i·15-s − 0.466·17-s + (−0.968 − 0.248i)19-s + 0.168i·21-s + 0.985·23-s − 0.213·25-s + 0.867i·27-s − 1.45i·29-s + 0.221i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(16.5668\)
Root analytic conductor: \(4.07023\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1),\ -0.248 + 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6627288842\)
\(L(\frac12)\) \(\approx\) \(0.6627288842\)
\(L(2)\) 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 \( 1 + (18.4 + 4.71i)T \)
good3 \( 1 - 1.48iT - 9T^{2} \)
5 \( 1 + 4.43T + 25T^{2} \)
7 \( 1 - 2.39T + 49T^{2} \)
11 \( 1 + 12.1T + 121T^{2} \)
13 \( 1 + 3.28iT - 169T^{2} \)
17 \( 1 + 7.93T + 289T^{2} \)
23 \( 1 - 22.6T + 529T^{2} \)
29 \( 1 + 42.2iT - 841T^{2} \)
31 \( 1 - 6.86iT - 961T^{2} \)
37 \( 1 + 40.8iT - 1.36e3T^{2} \)
41 \( 1 + 59.0iT - 1.68e3T^{2} \)
43 \( 1 + 1.93T + 1.84e3T^{2} \)
47 \( 1 + 12.5T + 2.20e3T^{2} \)
53 \( 1 + 89.9iT - 2.80e3T^{2} \)
59 \( 1 + 51.9iT - 3.48e3T^{2} \)
61 \( 1 - 14.1T + 3.72e3T^{2} \)
67 \( 1 - 74.7iT - 4.48e3T^{2} \)
71 \( 1 + 88.6iT - 5.04e3T^{2} \)
73 \( 1 - 8.33T + 5.32e3T^{2} \)
79 \( 1 + 7.24iT - 6.24e3T^{2} \)
83 \( 1 + 55.6T + 6.88e3T^{2} \)
89 \( 1 + 11.8iT - 7.92e3T^{2} \)
97 \( 1 - 20.8iT - 9.40e3T^{2} \)
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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

−10.36375101289870205671044772252, −9.351754576342856655400722581280, −8.325056541574291308210266585875, −7.65022835328243004685924530411, −6.74018100007768518779367383946, −5.33771848863779071396824331198, −4.49560835775105283250073810760, −3.65344805044551035974289394022, −2.22035080444620030928854391545, −0.25054723427883230467204399628, 1.44615467018311376872305746156, 2.85750303512464069234023165682, 4.21901980293026779460613239681, 4.98690022561005927290764787791, 6.39703532930893089938315678004, 7.26438307028740765167016038466, 7.955702034994338260005107147281, 8.705844391463069853866253897676, 9.935222639212620335689974181755, 10.82239810344177391468554881700

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