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Properties

Label 2-690-115.37-c1-0-8
Degree $2$
Conductor $690$
Sign $0.977 - 0.209i$
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  + (−0.877 − 0.479i)2-s + (0.997 + 0.0713i)3-s + (0.540 + 0.841i)4-s + (2.18 − 0.471i)5-s + (−0.841 − 0.540i)6-s + (1.33 + 3.57i)7-s + (−0.0713 − 0.997i)8-s + (0.989 + 0.142i)9-s + (−2.14 − 0.634i)10-s + (0.274 − 0.933i)11-s + (0.479 + 0.877i)12-s + (−0.363 − 0.135i)13-s + (0.543 − 3.77i)14-s + (2.21 − 0.314i)15-s + (−0.415 + 0.909i)16-s + (6.16 − 1.34i)17-s + ⋯
L(s)  = 1  + (−0.620 − 0.338i)2-s + (0.575 + 0.0411i)3-s + (0.270 + 0.420i)4-s + (0.977 − 0.210i)5-s + (−0.343 − 0.220i)6-s + (0.504 + 1.35i)7-s + (−0.0252 − 0.352i)8-s + (0.329 + 0.0474i)9-s + (−0.678 − 0.200i)10-s + (0.0826 − 0.281i)11-s + (0.138 + 0.253i)12-s + (−0.100 − 0.0375i)13-s + (0.145 − 1.00i)14-s + (0.571 − 0.0810i)15-s + (−0.103 + 0.227i)16-s + (1.49 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70630 + 0.180377i\)
\(L(\frac12)\) \(\approx\) \(1.70630 + 0.180377i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.877 + 0.479i)T \)
3 \( 1 + (-0.997 - 0.0713i)T \)
5 \( 1 + (-2.18 + 0.471i)T \)
23 \( 1 + (3.01 - 3.72i)T \)
good7 \( 1 + (-1.33 - 3.57i)T + (-5.29 + 4.58i)T^{2} \)
11 \( 1 + (-0.274 + 0.933i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (0.363 + 0.135i)T + (9.82 + 8.51i)T^{2} \)
17 \( 1 + (-6.16 + 1.34i)T + (15.4 - 7.06i)T^{2} \)
19 \( 1 + (4.73 - 3.04i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (1.75 - 2.72i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-0.0490 + 0.0566i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (5.11 + 6.83i)T + (-10.4 + 35.5i)T^{2} \)
41 \( 1 + (-0.842 - 5.86i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.138 + 1.93i)T + (-42.5 - 6.11i)T^{2} \)
47 \( 1 + (7.27 + 7.27i)T + 47iT^{2} \)
53 \( 1 + (-11.9 + 4.45i)T + (40.0 - 34.7i)T^{2} \)
59 \( 1 + (-11.1 + 5.08i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-5.42 - 4.69i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (2.25 - 4.12i)T + (-36.2 - 56.3i)T^{2} \)
71 \( 1 + (3.89 - 1.14i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-0.475 + 2.18i)T + (-66.4 - 30.3i)T^{2} \)
79 \( 1 + (-2.42 - 5.31i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-2.85 + 2.13i)T + (23.3 - 79.6i)T^{2} \)
89 \( 1 + (-0.830 - 0.958i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (10.6 + 7.97i)T + (27.3 + 93.0i)T^{2} \)
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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

−10.10774374884870044423922978058, −9.722236943496082809443970253484, −8.627646147574812590676837357406, −8.417581416064946329307472864985, −7.18899773553083051733097010519, −5.87878730250480660922614941174, −5.27881727209823295933670462951, −3.62985940054714500325972364413, −2.42101348310655705791403480402, −1.62912776054220637792590634305, 1.20818004079824413084679989149, 2.38227330769719283743071830736, 3.88443509501661899618233276322, 5.02715887987378633040304699992, 6.26699452040669738717653899937, 7.05262964377395511275271169453, 7.85513781970968259393933561176, 8.653318065959844401601324983462, 9.688566986032085148441491300748, 10.29816041811901842643366495596

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