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Properties

Label 2-357-7.4-c1-0-11
Degree $2$
Conductor $357$
Sign $0.909 - 0.416i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
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  + (0.614 + 1.06i)2-s + (0.5 − 0.866i)3-s + (0.243 − 0.422i)4-s + (1.20 + 2.09i)5-s + 1.22·6-s + (−1.25 − 2.33i)7-s + 3.05·8-s + (−0.499 − 0.866i)9-s + (−1.48 + 2.57i)10-s + (−1.73 + 3.01i)11-s + (−0.243 − 0.422i)12-s + 5.95·13-s + (1.71 − 2.76i)14-s + 2.41·15-s + (1.39 + 2.41i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.434 + 0.753i)2-s + (0.288 − 0.499i)3-s + (0.121 − 0.211i)4-s + (0.541 + 0.937i)5-s + 0.502·6-s + (−0.473 − 0.881i)7-s + 1.08·8-s + (−0.166 − 0.288i)9-s + (−0.470 + 0.814i)10-s + (−0.524 + 0.908i)11-s + (−0.0704 − 0.121i)12-s + 1.65·13-s + (0.457 − 0.739i)14-s + 0.624·15-s + (0.348 + 0.603i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05086 + 0.447132i\)
\(L(\frac12)\) \(\approx\) \(2.05086 + 0.447132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.25 + 2.33i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.614 - 1.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.20 - 2.09i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.73 - 3.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
19 \( 1 + (3.59 + 6.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.40 - 2.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.43T + 29T^{2} \)
31 \( 1 + (3.13 - 5.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.69T + 41T^{2} \)
43 \( 1 + 9.25T + 43T^{2} \)
47 \( 1 + (4.20 + 7.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.52 - 2.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.655 - 1.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.96 + 8.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.89 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.160 - 0.277i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (5.84 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.62T + 97T^{2} \)
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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

−11.23888484404335558000760103595, −10.62853844954227739945372226206, −9.852879634934567615844398589786, −8.489611002418854864482600629057, −7.23655673275969772442805967955, −6.76554477415657162323113124364, −6.07903243823837572757003819879, −4.69124216252260875302633814749, −3.27566833378330210661797642388, −1.73049599239753044564489853903, 1.79920648306069176825513120778, 3.13548690245999049701400976019, 4.06204631134957373758703573217, 5.40617544546839626085225088052, 6.17956126987413531041514765665, 8.079791138460556925270620288996, 8.659900725586000056785438637462, 9.568801347818736233410267248010, 10.70227088003005197371552378514, 11.32410916595284114744712178999

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