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Properties

Label 2-690-15.2-c1-0-31
Degree $2$
Conductor $690$
Sign $-0.749 + 0.662i$
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 + (−1.70 + 0.292i)3-s − 1.00i·4-s − 2.23i·5-s + (0.999 − 1.41i)6-s + (0.707 + 0.707i)8-s + (2.82 − i)9-s + (1.58 + 1.58i)10-s − 2.82i·11-s + (0.292 + 1.70i)12-s + (−2.16 + 2.16i)13-s + (0.654 + 3.81i)15-s − 1.00·16-s + (0.821 − 0.821i)17-s + (−1.29 + 2.70i)18-s + 1.16i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.985 + 0.169i)3-s − 0.500i·4-s − 0.999i·5-s + (0.408 − 0.577i)6-s + (0.250 + 0.250i)8-s + (0.942 − 0.333i)9-s + (0.500 + 0.500i)10-s − 0.852i·11-s + (0.0845 + 0.492i)12-s + (−0.599 + 0.599i)13-s + (0.169 + 0.985i)15-s − 0.250·16-s + (0.199 − 0.199i)17-s + (−0.304 + 0.638i)18-s + 0.266i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0966592 - 0.255443i\)
\(L(\frac12)\) \(\approx\) \(0.0966592 - 0.255443i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.292i)T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (2.16 - 2.16i)T - 13iT^{2} \)
17 \( 1 + (-0.821 + 0.821i)T - 17iT^{2} \)
19 \( 1 - 1.16iT - 19T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 + (1.16 + 1.16i)T + 37iT^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + (9.16 - 9.16i)T - 43iT^{2} \)
47 \( 1 + (0.229 - 0.229i)T - 47iT^{2} \)
53 \( 1 + (6.70 + 6.70i)T + 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (1.16 + 1.16i)T + 67iT^{2} \)
71 \( 1 - 5.88iT - 71T^{2} \)
73 \( 1 + (-5.32 + 5.32i)T - 73iT^{2} \)
79 \( 1 - 3.16iT - 79T^{2} \)
83 \( 1 + (3.05 + 3.05i)T + 83iT^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 + (5.16 + 5.16i)T + 97iT^{2} \)
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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

−9.943892597865109988203082231853, −9.343727406977813318527062591323, −8.457833127824509382019917848334, −7.48111739586740603870252541098, −6.53924592773266549912993185510, −5.56107998632482955061018887978, −4.99516117643223320313197563993, −3.83423178494857693586391526688, −1.62283979579580855374254019429, −0.19384124299261953454612110680, 1.73710811464688145499964739068, 3.00489455149613466222323655977, 4.33099613638906673084004542123, 5.44176989556629695622705457155, 6.57089358890826222174167168130, 7.27441533944002749296474951293, 7.995055506531091857585282902796, 9.520119625404438050413127259124, 10.07022353014594821858284131419, 10.81543787972614806773204465574

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