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Properties

Label 2-1368-1368.203-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.866 + 0.499i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−1.70 − 0.984i)11-s + (0.939 − 0.342i)12-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)18-s + (−0.766 − 0.642i)19-s + (1.26 + 1.50i)22-s − 24-s + (−0.766 − 0.642i)25-s − 0.999·27-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−1.70 − 0.984i)11-s + (0.939 − 0.342i)12-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)18-s + (−0.766 − 0.642i)19-s + (1.26 + 1.50i)22-s − 24-s + (−0.766 − 0.642i)25-s − 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.866 + 0.499i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5861709345\)
\(L(\frac12)\) \(\approx\) \(0.5861709345\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + 1.96iT - T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
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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

−9.150726187032358779225512205557, −8.679661450188846427457921439528, −7.83436041887298095709913020198, −7.44124281122044666527785703402, −6.35350497601198216286438898822, −5.60040544689009230958976090728, −3.93346284435676047901186448853, −2.76330371898249011864684337846, −2.24173585129427490355437791862, −0.57457456778215855691634205879, 2.03699543536134656839832326714, 2.83338289210601886688805026828, 4.25071783665429212909391063875, 5.21046010727415924409143605851, 5.94667955171198608780899742732, 7.24204171147477328841470679506, 7.916841557327709696445423456565, 8.405226869307436373210219853199, 9.603639352674730421743013692188, 9.773603757201584289164578102375

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