Properties

Label 2-690-115.7-c1-0-7
Degree $2$
Conductor $690$
Sign $0.628 - 0.778i$
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  + (0.997 − 0.0713i)2-s + (0.212 − 0.977i)3-s + (0.989 − 0.142i)4-s + (0.0285 + 2.23i)5-s + (0.142 − 0.989i)6-s + (−1.81 + 3.33i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (0.188 + 2.22i)10-s + (−0.824 + 0.714i)11-s + (0.0713 − 0.997i)12-s + (1.85 − 1.01i)13-s + (−1.57 + 3.45i)14-s + (2.19 + 0.447i)15-s + (0.959 − 0.281i)16-s + (4.77 + 6.37i)17-s + ⋯
L(s)  = 1  + (0.705 − 0.0504i)2-s + (0.122 − 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.0127 + 0.999i)5-s + (0.0580 − 0.404i)6-s + (−0.687 + 1.25i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.0594 + 0.704i)10-s + (−0.248 + 0.215i)11-s + (0.0205 − 0.287i)12-s + (0.514 − 0.280i)13-s + (−0.421 + 0.922i)14-s + (0.565 + 0.115i)15-s + (0.239 − 0.0704i)16-s + (1.15 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.628 - 0.778i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.628 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96647 + 0.939767i\)
\(L(\frac12)\) \(\approx\) \(1.96647 + 0.939767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.997 + 0.0713i)T \)
3 \( 1 + (-0.212 + 0.977i)T \)
5 \( 1 + (-0.0285 - 2.23i)T \)
23 \( 1 + (1.64 + 4.50i)T \)
good7 \( 1 + (1.81 - 3.33i)T + (-3.78 - 5.88i)T^{2} \)
11 \( 1 + (0.824 - 0.714i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (-1.85 + 1.01i)T + (7.02 - 10.9i)T^{2} \)
17 \( 1 + (-4.77 - 6.37i)T + (-4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.435 - 3.03i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-3.03 - 0.436i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-4.68 - 3.01i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (1.45 + 3.91i)T + (-27.9 + 24.2i)T^{2} \)
41 \( 1 + (3.28 + 7.19i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (7.16 + 1.55i)T + (39.1 + 17.8i)T^{2} \)
47 \( 1 + (-2.60 - 2.60i)T + 47iT^{2} \)
53 \( 1 + (-8.56 - 4.67i)T + (28.6 + 44.5i)T^{2} \)
59 \( 1 + (-2.33 + 7.94i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-2.89 + 4.51i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (0.00898 + 0.125i)T + (-66.3 + 9.53i)T^{2} \)
71 \( 1 + (-4.04 + 4.67i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-3.31 - 2.47i)T + (20.5 + 70.0i)T^{2} \)
79 \( 1 + (14.2 + 4.16i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (1.10 - 0.411i)T + (62.7 - 54.3i)T^{2} \)
89 \( 1 + (-1.56 + 1.00i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (6.93 + 2.58i)T + (73.3 + 63.5i)T^{2} \)
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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

−10.49669524904786198061465188201, −10.08698952337756945243819262514, −8.651140326432982773329493874164, −7.939253258223840871013312271245, −6.77388482374768230632128056380, −6.09784387690499537867275985535, −5.51200929460749520160763680461, −3.75552550145504379897538439360, −2.96332831335379091262616939522, −1.95610905242666198767199923700, 0.947488164410924271874879725463, 2.98850854609190719037658582621, 3.89726486800266967160737273348, 4.77215177064550890067176642009, 5.57209208617800320039078847600, 6.75951297793609889570051589630, 7.65481075565465759538657710795, 8.623138268176954014770427226363, 9.818518762658500445360374532064, 10.06342820266184831184712447924

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