Properties

Label 2-690-23.12-c1-0-2
Degree $2$
Conductor $690$
Sign $0.726 - 0.687i$
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 + (1.76 + 2.03i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (1.98 + 1.27i)11-s + (0.841 + 0.540i)12-s + (−1.36 + 1.57i)13-s + (2.57 + 0.757i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (1.96 + 4.30i)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 + (0.665 + 0.767i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.207 + 0.238i)10-s + (0.597 + 0.383i)11-s + (0.242 + 0.156i)12-s + (−0.377 + 0.435i)13-s + (0.689 + 0.202i)14-s + (−0.0367 − 0.255i)15-s + (−0.163 − 0.188i)16-s + (0.477 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.726 - 0.687i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85789 + 0.740162i\)
\(L(\frac12)\) \(\approx\) \(1.85789 + 0.740162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.956 - 4.69i)T \)
good7 \( 1 + (-1.76 - 2.03i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-1.98 - 1.27i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.36 - 1.57i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.96 - 4.30i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-1.47 + 3.22i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.69 - 5.89i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.280 + 1.95i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-7.79 - 2.28i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (0.914 - 0.268i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.512 + 3.56i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 1.28T + 47T^{2} \)
53 \( 1 + (2.57 + 2.96i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (0.927 - 1.07i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.359 + 2.50i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (4.29 - 2.76i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-0.461 + 0.296i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-3.91 + 8.56i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-0.555 + 0.640i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (6.18 + 1.81i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.65 + 11.4i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-2.77 + 0.813i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.75920061230020423507116667756, −9.778693491434929679131545855262, −9.026780119011472703385413401090, −8.041608944155772069196276010716, −6.93113575061999475747321946236, −5.83814042297034599087286703404, −4.94703690782182897997700348398, −4.13367969898718085741563099152, −3.07007526111445141684424021472, −1.70950236440145570755051958570, 0.975369413382219377401812385896, 2.73036620996674859292893676953, 3.98464656614349886168231553659, 4.85422630345252956542339760703, 5.91998567185297687456561911872, 6.86694713645471880210157404544, 7.76893809367344533318539096221, 8.139710377901481256905222730758, 9.437946413057009622905133268572, 10.56976825762457164343917609041

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