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Properties

Label 2-686-49.39-c1-0-13
Degree $2$
Conductor $686$
Sign $0.950 - 0.310i$
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 + (−0.573 + 0.391i)3-s + (0.955 + 0.294i)4-s + (−0.137 − 1.83i)5-s + (−0.625 + 0.301i)6-s + (0.900 + 0.433i)8-s + (−0.919 + 2.34i)9-s + (0.137 − 1.83i)10-s + (1.46 + 3.72i)11-s + (−0.663 + 0.204i)12-s + (3.60 − 4.52i)13-s + (0.797 + 0.999i)15-s + (0.826 + 0.563i)16-s + (2.53 + 2.35i)17-s + (−1.25 + 2.18i)18-s + (1.51 + 2.63i)19-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (−0.331 + 0.225i)3-s + (0.477 + 0.147i)4-s + (−0.0615 − 0.820i)5-s + (−0.255 + 0.123i)6-s + (0.318 + 0.153i)8-s + (−0.306 + 0.781i)9-s + (0.0434 − 0.580i)10-s + (0.440 + 1.12i)11-s + (−0.191 + 0.0591i)12-s + (1.00 − 1.25i)13-s + (0.205 + 0.258i)15-s + (0.206 + 0.140i)16-s + (0.614 + 0.570i)17-s + (−0.296 + 0.513i)18-s + (0.348 + 0.603i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(686\)    =    \(2 \cdot 7^{3}\)
Sign: $0.950 - 0.310i$
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: \((2,\ 686,\ (\ :1/2),\ 0.950 - 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14904 + 0.341549i\)
\(L(\frac12)\) \(\approx\) \(2.14904 + 0.341549i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.573 - 0.391i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.137 + 1.83i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-1.46 - 3.72i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-3.60 + 4.52i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-2.53 - 2.35i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-1.51 - 2.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.90 + 5.48i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.392 + 1.72i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (2.70 - 4.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.88 + 1.81i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (2.82 + 1.36i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.50 - 1.68i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (7.20 + 1.08i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (8.55 + 2.63i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.537 - 7.16i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-3.69 + 1.14i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (0.927 - 1.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.61 - 7.06i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (11.5 - 1.74i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-2.23 - 3.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.42 - 4.29i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.17 - 10.6i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 9.72T + 97T^{2} \)
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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

−10.63397440008135989579011259814, −9.861812443978355535592221382314, −8.560712286092091486687404601231, −8.000955624598222947373219019849, −6.84227086404004727131062965517, −5.73849733097613141902430445025, −5.07288455652985928229946564382, −4.25233382100156203932743078805, −3.01312923547431807529714441169, −1.39030291961547542765320147118, 1.24172512248319908914746249430, 3.09890778885944578655880127691, 3.58758026815484327168534175857, 5.03239865164087685134554108742, 6.15439428454196439105698197449, 6.58387654422684717638555042188, 7.48645074823636407372192997497, 8.874415740413876735988740116557, 9.515896612612577838137523132015, 10.91964379242191672947825443173

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