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Properties

Label 2-690-345.344-c1-0-8
Degree $2$
Conductor $690$
Sign $0.576 - 0.816i$
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  − 2-s + (−0.250 − 1.71i)3-s + 4-s + (0.545 + 2.16i)5-s + (0.250 + 1.71i)6-s − 1.86·7-s − 8-s + (−2.87 + 0.858i)9-s + (−0.545 − 2.16i)10-s + 2.81·11-s + (−0.250 − 1.71i)12-s − 1.42i·13-s + 1.86·14-s + (3.58 − 1.47i)15-s + 16-s + 3.10i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.144 − 0.989i)3-s + 0.5·4-s + (0.243 + 0.969i)5-s + (0.102 + 0.699i)6-s − 0.703·7-s − 0.353·8-s + (−0.958 + 0.286i)9-s + (−0.172 − 0.685i)10-s + 0.849·11-s + (−0.0723 − 0.494i)12-s − 0.395i·13-s + 0.497·14-s + (0.924 − 0.381i)15-s + 0.250·16-s + 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.576 - 0.816i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.576 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703098 + 0.364272i\)
\(L(\frac12)\) \(\approx\) \(0.703098 + 0.364272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.250 + 1.71i)T \)
5 \( 1 + (-0.545 - 2.16i)T \)
23 \( 1 + (4.05 - 2.56i)T \)
good7 \( 1 + 1.86T + 7T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 + 1.42iT - 13T^{2} \)
17 \( 1 - 3.10iT - 17T^{2} \)
19 \( 1 - 3.42iT - 19T^{2} \)
29 \( 1 - 6.01iT - 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 - 1.27T + 37T^{2} \)
41 \( 1 - 1.76iT - 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 - 2.70T + 47T^{2} \)
53 \( 1 - 5.16iT - 53T^{2} \)
59 \( 1 - 5.36iT - 59T^{2} \)
61 \( 1 - 6.07iT - 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 - 4.56iT - 71T^{2} \)
73 \( 1 + 1.66iT - 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 - 6.08iT - 83T^{2} \)
89 \( 1 + 8.61T + 89T^{2} \)
97 \( 1 + 7.72T + 97T^{2} \)
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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

−10.52714001593803337433891525980, −9.818813296003014530201435495075, −8.822220420402412919405800807475, −7.86214089212138297118312345690, −7.12186477413881637736224317299, −6.27540984699902999549683776398, −5.83644084062197371149596929358, −3.69495334715768472396094670346, −2.64866258131949916725095719890, −1.40967846158744922476354208491, 0.56025911053008423830498012646, 2.45665558157916700507924024293, 3.88445500905009872073773002919, 4.76346888776898156168990146272, 5.93183815033305558140058867963, 6.67580766050390283583989762282, 8.079580907962778528687707352455, 8.908906850733701694546891092987, 9.565564351356103889938431711180, 9.901970104657231292009159182243

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