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Properties

Label 2-2394-21.5-c1-0-16
Degree $2$
Conductor $2394$
Sign $0.291 - 0.956i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.566 + 0.981i)5-s + (−0.203 + 2.63i)7-s − 0.999i·8-s + (0.981 − 0.566i)10-s + (2.97 − 1.71i)11-s + 1.17i·13-s + (1.49 − 2.18i)14-s + (−0.5 + 0.866i)16-s + (−0.471 − 0.817i)17-s + (0.866 + 0.5i)19-s − 1.13·20-s − 3.43·22-s + (3.02 + 1.74i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.253 + 0.438i)5-s + (−0.0769 + 0.997i)7-s − 0.353i·8-s + (0.310 − 0.179i)10-s + (0.896 − 0.517i)11-s + 0.326i·13-s + (0.399 − 0.583i)14-s + (−0.125 + 0.216i)16-s + (−0.114 − 0.198i)17-s + (0.198 + 0.114i)19-s − 0.253·20-s − 0.731·22-s + (0.631 + 0.364i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.291 - 0.956i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (2357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.291 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160789046\)
\(L(\frac12)\) \(\approx\) \(1.160789046\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.203 - 2.63i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (0.566 - 0.981i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.97 + 1.71i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.17iT - 13T^{2} \)
17 \( 1 + (0.471 + 0.817i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.02 - 1.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.191iT - 29T^{2} \)
31 \( 1 + (3.45 - 1.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 + 2.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + (-5.74 + 9.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.09 - 3.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.890 + 1.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.52 + 4.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.96 - 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.68iT - 71T^{2} \)
73 \( 1 + (4.41 - 2.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.39 - 7.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-1.75 + 3.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.99iT - 97T^{2} \)
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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

−9.060933557598902089811618886406, −8.664926086342799173490029869612, −7.59125834940267504578565461369, −6.96038801560436404827007006088, −6.10866140214169668968785403896, −5.30235660966571432069626802932, −4.04999584417952785264564964265, −3.22192584858395843698759228404, −2.36910882583362723580115437932, −1.17354494934459513218392727106, 0.56598732576400602542016263534, 1.53276764147172218257641283573, 2.96245394492760372657509309509, 4.19689321240767180409817818367, 4.66575915236916553423100389246, 5.91414672794206858120470223315, 6.62170196365520978780492446113, 7.43435313139285391437571300891, 7.898720581156773593061863422491, 8.924241450965766973700241924496

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