Properties

Label 2-690-23.13-c1-0-7
Degree $2$
Conductor $690$
Sign $-0.342 - 0.939i$
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  + (0.654 + 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (0.142 + 0.989i)6-s + (1.94 + 0.571i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (−3.23 + 3.72i)11-s + (−0.654 + 0.755i)12-s + (6.08 − 1.78i)13-s + (0.842 + 1.84i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (−0.738 − 5.13i)17-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.185 + 0.406i)5-s + (0.0580 + 0.404i)6-s + (0.735 + 0.215i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.303 + 0.0890i)10-s + (−0.974 + 1.12i)11-s + (−0.189 + 0.218i)12-s + (1.68 − 0.495i)13-s + (0.225 + 0.493i)14-s + (−0.217 + 0.139i)15-s + (−0.239 − 0.0704i)16-s + (−0.179 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27870 + 1.82766i\)
\(L(\frac12)\) \(\approx\) \(1.27870 + 1.82766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (4.78 - 0.251i)T \)
good7 \( 1 + (-1.94 - 0.571i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (3.23 - 3.72i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-6.08 + 1.78i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.738 + 5.13i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.914 - 6.36i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-0.160 - 1.11i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-1.29 + 0.830i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-3.68 - 8.07i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-2.21 + 4.86i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (6.99 + 4.49i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + (-2.01 - 0.592i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-1.34 + 0.395i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-0.181 + 0.116i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-2.92 - 3.37i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (10.1 + 11.6i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.620 + 4.31i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-14.8 + 4.34i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-0.0838 - 0.183i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-7.42 - 4.77i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-5.87 + 12.8i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.62018282536579476308140404576, −9.969290058631204212517471876782, −8.723579069372898741169131354117, −8.004180468111523975247698280248, −7.43500338772929201148662635525, −6.20087728315751631571599509216, −5.26229101338117868858285326261, −4.31071645380084629511983006552, −3.31937684564009163263856431694, −2.06272089511624428102981755127, 1.04282386824793489388567903815, 2.33234853031390037170099802044, 3.63578519776724854948024948367, 4.41704539105524830027363305192, 5.66373865444993290475984149386, 6.45417052300266017633520652596, 7.913397781287490367696503282565, 8.442425374289864831628661178042, 9.173188417781643479322224177196, 10.59443126604660752283319148053

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