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Properties

Label 2-608-19.18-c2-0-26
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

\(L(\frac{3}{2})\) \(\approx\) \(1.267478154\)
\(L(\frac12)\) \(\approx\) \(1.267478154\)
\(L(2)\) 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

−10.06692122469431409739014707101, −9.397290199968407616913580028353, −8.235233484626881230276565647605, −7.50397581438897112803937436586, −6.74659545311648512964131749368, −5.77507271325719062396131653603, −4.25510652568701711861020716336, −3.66586303707008178684698063886, −1.99865985893481326120369677561, −0.51963538450747291883232204418, 1.37198133088595991435255823281, 3.26058539348311106367074380940, 4.07246619175203240204032511698, 4.88174211268247061313148320212, 6.34088960083639821709628359158, 7.11551768922271361363390755322, 8.040670908066725364546575359746, 9.142491789069016945109802910883, 9.712375638929093633424258129484, 10.69059506506444174803436819855

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