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Properties

Label 2-690-15.8-c1-0-31
Degree $2$
Conductor $690$
Sign $0.800 - 0.599i$
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  + (0.707 + 0.707i)2-s + (1.60 − 0.639i)3-s + 1.00i·4-s + (1.63 + 1.52i)5-s + (1.59 + 0.685i)6-s + (0.772 − 0.772i)7-s + (−0.707 + 0.707i)8-s + (2.18 − 2.05i)9-s + (0.0731 + 2.23i)10-s − 1.36i·11-s + (0.639 + 1.60i)12-s + (0.808 + 0.808i)13-s + 1.09·14-s + (3.60 + 1.41i)15-s − 1.00·16-s + (−2.66 − 2.66i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.929 − 0.369i)3-s + 0.500i·4-s + (0.729 + 0.683i)5-s + (0.649 + 0.279i)6-s + (0.291 − 0.291i)7-s + (−0.250 + 0.250i)8-s + (0.727 − 0.686i)9-s + (0.0231 + 0.706i)10-s − 0.411i·11-s + (0.184 + 0.464i)12-s + (0.224 + 0.224i)13-s + 0.291·14-s + (0.930 + 0.365i)15-s − 0.250·16-s + (−0.645 − 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85585 + 0.950839i\)
\(L(\frac12)\) \(\approx\) \(2.85585 + 0.950839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.60 + 0.639i)T \)
5 \( 1 + (-1.63 - 1.52i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-0.772 + 0.772i)T - 7iT^{2} \)
11 \( 1 + 1.36iT - 11T^{2} \)
13 \( 1 + (-0.808 - 0.808i)T + 13iT^{2} \)
17 \( 1 + (2.66 + 2.66i)T + 17iT^{2} \)
19 \( 1 - 1.89iT - 19T^{2} \)
29 \( 1 - 1.81T + 29T^{2} \)
31 \( 1 + 5.22T + 31T^{2} \)
37 \( 1 + (7.05 - 7.05i)T - 37iT^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-4.99 - 4.99i)T + 43iT^{2} \)
47 \( 1 + (7.28 + 7.28i)T + 47iT^{2} \)
53 \( 1 + (-1.52 + 1.52i)T - 53iT^{2} \)
59 \( 1 - 5.33T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (-8.07 + 8.07i)T - 67iT^{2} \)
71 \( 1 + 2.85iT - 71T^{2} \)
73 \( 1 + (-8.07 - 8.07i)T + 73iT^{2} \)
79 \( 1 + 4.48iT - 79T^{2} \)
83 \( 1 + (-0.994 + 0.994i)T - 83iT^{2} \)
89 \( 1 + 3.62T + 89T^{2} \)
97 \( 1 + (0.314 - 0.314i)T - 97iT^{2} \)
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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

−10.50774120633922082419443223273, −9.514024946491207621209258827585, −8.737953130255844835256699416486, −7.80984736122436013763045521482, −6.96474950432231704725264594884, −6.33929991098162151058444581664, −5.17600984917955499864302510260, −3.88965703031307836127482189656, −2.96344296792773421364226033266, −1.80338837680355960816918943956, 1.65046029184048056496399012737, 2.51059087696502408265556749226, 3.84473125828153591183917066693, 4.74193144217355779758213534108, 5.56325621245298519115890850906, 6.76755638404419792022342131298, 8.056804756645744802501405361693, 8.881168381516293399572993594684, 9.461801134786851796241914622458, 10.36137594243894973940380785056

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