Properties

Label 2-690-23.3-c1-0-11
Degree $2$
Conductor $690$
Sign $0.391 + 0.920i$
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  + (0.142 + 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (−0.959 − 0.281i)6-s + (−2.22 + 1.43i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (0.856 − 5.95i)11-s + (0.142 − 0.989i)12-s + (−5.09 − 3.27i)13-s + (−1.73 − 2.00i)14-s + (0.415 + 0.909i)15-s + (0.841 − 0.540i)16-s + (−1.34 − 0.395i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.292 − 0.337i)5-s + (−0.391 − 0.115i)6-s + (−0.841 + 0.540i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.266 + 0.170i)10-s + (0.258 − 1.79i)11-s + (0.0410 − 0.285i)12-s + (−1.41 − 0.908i)13-s + (−0.463 − 0.534i)14-s + (0.107 + 0.234i)15-s + (0.210 − 0.135i)16-s + (−0.326 − 0.0959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473600 - 0.313311i\)
\(L(\frac12)\) \(\approx\) \(0.473600 - 0.313311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + (0.415 - 0.909i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (4.00 - 2.63i)T \)
good7 \( 1 + (2.22 - 1.43i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.856 + 5.95i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (5.09 + 3.27i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (1.34 + 0.395i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-1.44 + 0.424i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-1.48 - 0.436i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.799 - 1.75i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (5.68 + 6.56i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-1.73 + 2.00i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-3.80 + 8.33i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 0.964T + 47T^{2} \)
53 \( 1 + (6.68 - 4.29i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-5.92 - 3.80i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (6.02 + 13.1i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (0.0565 + 0.393i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-1.49 - 10.4i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-5.18 + 1.52i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-10.1 - 6.52i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (7.80 + 9.00i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (0.651 - 1.42i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (6.18 - 7.14i)T + (-13.8 - 96.0i)T^{2} \)
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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

−10.10298356222080448010137598605, −9.335525431592162084105092613753, −8.704558551467356509513109477785, −7.71781292517322653268797560149, −6.53835917609314350230615292197, −5.67829992697758662392842512192, −5.22408329930894639014819722493, −3.79306198767572738414153403986, −2.81044625451680112369598553587, −0.28439735182725302788735873648, 1.76535170495995472841063842934, 2.68494336897341343293954173067, 4.16379350701685473557350582781, 4.93710105504606850645241045476, 6.45059092790516067343222587425, 6.94669852411288200657392076610, 7.86856733541841039611173381670, 9.410678707013326726058283804041, 9.805727810874638438221952781332, 10.47833208977272498889245207695

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