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Properties

Label 2-686-49.39-c1-0-22
Degree $2$
Conductor $686$
Sign $0.346 + 0.937i$
Analytic cond. $5.47773$
Root an. cond. $2.34045$
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  + (0.988 + 0.149i)2-s + (1.98 − 1.35i)3-s + (0.955 + 0.294i)4-s + (−0.295 − 3.94i)5-s + (2.16 − 1.04i)6-s + (0.900 + 0.433i)8-s + (1.00 − 2.57i)9-s + (0.295 − 3.94i)10-s + (1.44 + 3.68i)11-s + (2.29 − 0.707i)12-s + (−1.13 + 1.42i)13-s + (−5.92 − 7.43i)15-s + (0.826 + 0.563i)16-s + (−0.951 − 0.882i)17-s + (1.38 − 2.39i)18-s + (1.57 + 2.73i)19-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (1.14 − 0.780i)3-s + (0.477 + 0.147i)4-s + (−0.132 − 1.76i)5-s + (0.882 − 0.425i)6-s + (0.318 + 0.153i)8-s + (0.336 − 0.857i)9-s + (0.0935 − 1.24i)10-s + (0.436 + 1.11i)11-s + (0.662 − 0.204i)12-s + (−0.314 + 0.394i)13-s + (−1.52 − 1.91i)15-s + (0.206 + 0.140i)16-s + (−0.230 − 0.214i)17-s + (0.325 − 0.563i)18-s + (0.361 + 0.626i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(686\)    =    \(2 \cdot 7^{3}\)
Sign: $0.346 + 0.937i$
Analytic conductor: \(5.47773\)
Root analytic conductor: \(2.34045\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{686} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 686,\ (\ :1/2),\ 0.346 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61729 - 1.82251i\)
\(L(\frac12)\) \(\approx\) \(2.61729 - 1.82251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 \)
good3 \( 1 + (-1.98 + 1.35i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.295 + 3.94i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-1.44 - 3.68i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (1.13 - 1.42i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.951 + 0.882i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-1.57 - 2.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.50 - 1.40i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.60 + 7.04i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.97 + 3.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.41 - 0.435i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (8.00 + 3.85i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (5.80 - 2.79i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-6.48 - 0.977i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-9.75 - 3.00i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (1.00 - 13.4i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (5.41 - 1.67i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.482 + 0.835i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.18 - 5.21i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-4.04 + 0.609i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-6.03 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.791 - 0.992i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-4.72 + 12.0i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−9.994562115304752354498392471326, −9.239982275063097774791704499298, −8.450176285503993501031192264879, −7.74599639576594028477469941556, −6.97476596445732401444455894259, −5.67370486060956850764878659247, −4.62196092770885644600419467336, −3.91552643982090552028451646062, −2.33519460908627271986041008053, −1.42240375256200739938928987475, 2.38533367610195499613273042369, 3.32733882974588193552660008568, 3.56200756849761667946192437764, 4.99049033687345196072215901898, 6.31558404446389406388898599904, 6.98146771263651762015142466717, 8.070265598265981115877608632617, 8.931991513461878470294803785818, 10.04715175695342643256980348395, 10.58563998099012736862661880398

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