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Properties

Label 2-2394-21.20-c1-0-13
Degree $2$
Conductor $2394$
Sign $-0.139 - 0.990i$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
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  + i·2-s − 4-s + 0.335·5-s + (−2.35 − 1.21i)7-s − i·8-s + 0.335i·10-s − 0.185i·11-s − 0.757i·13-s + (1.21 − 2.35i)14-s + 16-s − 0.456·17-s + i·19-s − 0.335·20-s + 0.185·22-s − 1.64i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.149·5-s + (−0.889 − 0.457i)7-s − 0.353i·8-s + 0.106i·10-s − 0.0558i·11-s − 0.209i·13-s + (0.323 − 0.628i)14-s + 0.250·16-s − 0.110·17-s + 0.229i·19-s − 0.0749·20-s + 0.0394·22-s − 0.344i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.222715324\)
\(L(\frac12)\) \(\approx\) \(1.222715324\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (2.35 + 1.21i)T \)
19 \( 1 - iT \)
good5 \( 1 - 0.335T + 5T^{2} \)
11 \( 1 + 0.185iT - 11T^{2} \)
13 \( 1 + 0.757iT - 13T^{2} \)
17 \( 1 + 0.456T + 17T^{2} \)
23 \( 1 + 1.64iT - 23T^{2} \)
29 \( 1 - 3.90iT - 29T^{2} \)
31 \( 1 - 9.21iT - 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 5.93T + 43T^{2} \)
47 \( 1 - 2.36T + 47T^{2} \)
53 \( 1 + 4.49iT - 53T^{2} \)
59 \( 1 + 6.16T + 59T^{2} \)
61 \( 1 - 4.83iT - 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 5.10iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 - 0.519T + 79T^{2} \)
83 \( 1 + 8.81T + 83T^{2} \)
89 \( 1 - 3.20T + 89T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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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

−9.164708823858847294245262874919, −8.341214616282868042184769506160, −7.51452554878737376752513772734, −6.87162876481071481104946224125, −6.12110587163224442600679132396, −5.45666537646277859132277128002, −4.38852824700748252366664885972, −3.62532092993358217378024621989, −2.59098367900698328805426155849, −0.975860479366878436994642999968, 0.51084845386898621225981509869, 2.06756055587579420842823065205, 2.76780878020860300232871284868, 3.83415315107174406086996342788, 4.52292691913765118502622220364, 5.84260722763897908181032011847, 6.08896407853453438658090659829, 7.35637025233495998074026908666, 8.029739875826726971494934485083, 9.153896108632118431281695533022

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