Properties

Label 2-690-23.13-c1-0-2
Degree $2$
Conductor $690$
Sign $-0.991 - 0.133i$
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  + (0.654 + 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (0.142 + 0.989i)6-s + (−4.28 − 1.25i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.0345 − 0.0398i)11-s + (−0.654 + 0.755i)12-s + (−4.11 + 1.20i)13-s + (−1.85 − 4.06i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.686 + 4.77i)17-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.185 + 0.406i)5-s + (0.0580 + 0.404i)6-s + (−1.62 − 0.475i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.303 + 0.0890i)10-s + (0.0104 − 0.0120i)11-s + (−0.189 + 0.218i)12-s + (−1.14 + 0.335i)13-s + (−0.496 − 1.08i)14-s + (−0.217 + 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.166 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.991 - 0.133i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0762740 + 1.13583i\)
\(L(\frac12)\) \(\approx\) \(0.0762740 + 1.13583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (2.00 - 4.35i)T \)
good7 \( 1 + (4.28 + 1.25i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-0.0345 + 0.0398i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (4.11 - 1.20i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.686 - 4.77i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.891 - 6.20i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.662 + 4.60i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.42 + 1.55i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (0.733 + 1.60i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-4.24 + 9.29i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-9.31 - 5.98i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 2.02T + 47T^{2} \)
53 \( 1 + (-1.07 - 0.315i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (6.35 - 1.86i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-6.79 + 4.36i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-7.19 - 8.30i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-6.02 - 6.95i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.56 + 10.8i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (11.3 - 3.33i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (5.27 + 11.5i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-10.3 - 6.64i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (5.80 - 12.7i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.61915604507132484272847392303, −9.906375352026049825978536176542, −9.319215646335517263856260313184, −8.016653200009365108107018834369, −7.38790356656677248385352069153, −6.41776391031616407302391616156, −5.69578184174953144239803758738, −4.08358426115384729789720245097, −3.66637502578043250875403357895, −2.44980942981018973707605447869, 0.45434647084272503905863311670, 2.56096615689363926783856688868, 3.02350056033479632746448811094, 4.40555692232424913556559498888, 5.38448730395550386659411691946, 6.56767849000555186730130399404, 7.22945411621778188865185162509, 8.551203994036400305974744814147, 9.450631282323944332764793552527, 9.786620515397632858448072199959

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