Properties

Label 2-690-23.4-c1-0-4
Degree $2$
Conductor $690$
Sign $0.987 - 0.155i$
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  + (−0.415 − 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.654 + 0.755i)6-s + (−0.0867 − 0.603i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (−0.0982 + 0.215i)11-s + (0.415 − 0.909i)12-s + (−0.227 + 1.58i)13-s + (−0.512 + 0.329i)14-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (0.744 + 0.858i)17-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.376 + 0.241i)5-s + (0.267 + 0.308i)6-s + (−0.0327 − 0.227i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0450 − 0.313i)10-s + (−0.0296 + 0.0648i)11-s + (0.119 − 0.262i)12-s + (−0.0632 + 0.439i)13-s + (−0.136 + 0.0880i)14-s + (−0.247 − 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (0.180 + 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05703 + 0.0826604i\)
\(L(\frac12)\) \(\approx\) \(1.05703 + 0.0826604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (1.17 - 4.64i)T \)
good7 \( 1 + (0.0867 + 0.603i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.0982 - 0.215i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.227 - 1.58i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-0.744 - 0.858i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-0.557 + 0.643i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-2.69 - 3.11i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-8.03 - 2.35i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-3.81 + 2.44i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-6.93 - 4.45i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (0.394 - 0.115i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + (-1.01 - 7.04i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (0.0718 - 0.499i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-11.5 - 3.39i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (1.41 + 3.08i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-0.611 - 1.34i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-3.10 + 3.58i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.829 + 5.76i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (3.06 - 1.96i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (3.23 - 0.949i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (0.0626 + 0.0402i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.45127587830629542837045087817, −9.830735396430138025264280760753, −9.050846655154091225094948049056, −7.941883286546686580460743233821, −6.96442787532891675060301361799, −6.00845358911051948230244684289, −4.91815686744399947316198651120, −3.90843726146132528892940652542, −2.66518974701670702770820826229, −1.22130242111362098922390162490, 0.799058489294547117754411569997, 2.50315895157652616875735125805, 4.26091012438415823078933904670, 5.24556297492415221949924741315, 6.04517713356981357865936324357, 6.77760425584063999786440886304, 7.898073141116427006700441236513, 8.548880441640747000037568327625, 9.691904126099724175841253172965, 10.20456258454845384443509190962

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