Properties

Label 2-690-23.2-c1-0-11
Degree $2$
Conductor $690$
Sign $0.855 + 0.517i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (1.26 − 1.46i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (4.28 − 2.75i)11-s + (−0.841 + 0.540i)12-s + (−1.52 − 1.75i)13-s + (−1.85 + 0.546i)14-s + (−0.142 + 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.233 − 0.512i)17-s + ⋯
L(s)  = 1  + (−0.594 − 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (0.169 − 0.371i)6-s + (0.479 − 0.553i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (1.29 − 0.830i)11-s + (−0.242 + 0.156i)12-s + (−0.422 − 0.487i)13-s + (−0.496 + 0.145i)14-s + (−0.0367 + 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.0567 − 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29912 - 0.362557i\)
\(L(\frac12)\) \(\approx\) \(1.29912 - 0.362557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-1.93 - 4.38i)T \)
good7 \( 1 + (-1.26 + 1.46i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-4.28 + 2.75i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (1.52 + 1.75i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-0.233 + 0.512i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (2.44 + 5.35i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-2.35 + 5.16i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.222 - 1.54i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (2.16 - 0.636i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-11.2 - 3.31i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (0.508 + 3.53i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 + (3.60 - 4.15i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-4.61 - 5.32i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-1.35 + 9.42i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-4.95 - 3.18i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-1.55 - 0.998i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (1.85 + 4.05i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-3.29 - 3.80i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-1.14 + 0.335i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.43 + 9.96i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (13.2 + 3.88i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.40810825598622945820491022276, −9.461690369865735222794941279514, −8.962437915440130585022079831993, −7.969940401560096393542284781608, −6.98642002261151326295613908071, −5.97627008521469595636493786602, −4.72661233978638137136748551004, −3.72362011552374322145877938818, −2.58759971087784423790224781400, −0.998347574969434158164411272927, 1.42850810231031640454503826952, 2.31246571400818632285937281788, 4.16549931291690295593336549446, 5.33440572969211468291029845512, 6.37011417915091903158497229602, 6.96027409825631173481874480486, 8.018204993004576533794805982219, 8.824861795096710508591204819815, 9.446636933136740134572833801297, 10.37531249762780637676341443860

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