Properties

Label 2-690-23.16-c1-0-9
Degree $2$
Conductor $690$
Sign $0.393 + 0.919i$
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 + (4.58 − 1.34i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (0.959 + 0.281i)10-s + (−0.612 − 0.706i)11-s + (0.654 + 0.755i)12-s + (−6.39 − 1.87i)13-s + (−1.98 + 4.35i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (−0.0823 + 0.572i)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 + (1.73 − 0.509i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (0.303 + 0.0890i)10-s + (−0.184 − 0.212i)11-s + (0.189 + 0.218i)12-s + (−1.77 − 0.520i)13-s + (−0.531 + 1.16i)14-s + (0.217 + 0.139i)15-s + (−0.239 + 0.0704i)16-s + (−0.0199 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.393 + 0.919i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656040 - 0.432704i\)
\(L(\frac12)\) \(\approx\) \(0.656040 - 0.432704i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.66 - 3.09i)T \)
good7 \( 1 + (-4.58 + 1.34i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (0.612 + 0.706i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (6.39 + 1.87i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.0823 - 0.572i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (1.08 + 7.57i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (0.183 - 1.27i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (2.87 + 1.84i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-1.16 + 2.55i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (3.53 + 7.73i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-7.18 + 4.61i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 1.94T + 47T^{2} \)
53 \( 1 + (-9.10 + 2.67i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (10.8 + 3.19i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (3.05 + 1.96i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-6.31 + 7.28i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (6.65 - 7.67i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.83 - 12.7i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-4.08 - 1.20i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (1.13 - 2.48i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-2.33 + 1.50i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (2.27 + 4.97i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.37057122420717195518819899968, −9.372944923366983955364325987797, −8.514793607288011988629175205092, −7.57373242892595458743295445266, −7.15401438126972810955177223097, −5.52753308291925794842169540262, −5.01041269951764340386394674498, −4.20978809082864845414411626367, −2.15011225247625502358484927217, −0.51108007355983919765827958533, 1.67195088676750891971185002243, 2.48640942018391993327470387263, 4.28209842255176804277200459539, 5.02874424023390773301322175505, 6.19794296852128631852779646565, 7.64929865018141740869640689777, 7.73954927063513981969849914204, 8.894128931946445363337479399838, 10.02897283362400881468168861288, 10.62201611620838502162694382375

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