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Properties

Label 2-357-119.16-c1-0-8
Degree $2$
Conductor $357$
Sign $0.999 + 0.0142i$
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  + (−0.636 + 1.10i)2-s + (−0.866 + 0.5i)3-s + (0.188 + 0.326i)4-s + (−2.74 − 1.58i)5-s − 1.27i·6-s + (1.56 − 2.13i)7-s − 3.02·8-s + (0.499 − 0.866i)9-s + (3.50 − 2.02i)10-s + (4.27 − 2.47i)11-s + (−0.326 − 0.188i)12-s + 0.743·13-s + (1.35 + 3.08i)14-s + 3.17·15-s + (1.55 − 2.68i)16-s + (−1.94 + 3.63i)17-s + ⋯
L(s)  = 1  + (−0.450 + 0.780i)2-s + (−0.499 + 0.288i)3-s + (0.0943 + 0.163i)4-s + (−1.22 − 0.709i)5-s − 0.520i·6-s + (0.591 − 0.806i)7-s − 1.07·8-s + (0.166 − 0.288i)9-s + (1.10 − 0.639i)10-s + (1.29 − 0.744i)11-s + (−0.0943 − 0.0544i)12-s + 0.206·13-s + (0.362 + 0.824i)14-s + 0.819·15-s + (0.387 − 0.671i)16-s + (−0.472 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.999 + 0.0142i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.999 + 0.0142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754703 - 0.00537503i\)
\(L(\frac12)\) \(\approx\) \(0.754703 - 0.00537503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-1.56 + 2.13i)T \)
17 \( 1 + (1.94 - 3.63i)T \)
good2 \( 1 + (0.636 - 1.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.74 + 1.58i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.27 + 2.47i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.743T + 13T^{2} \)
19 \( 1 + (-3.14 + 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.82 - 1.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.08iT - 29T^{2} \)
31 \( 1 + (4.76 - 2.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.27 - 1.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 - 8.93T + 43T^{2} \)
47 \( 1 + (-4.61 + 7.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.57 - 6.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.996 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.35 + 1.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.24 + 9.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.26iT - 71T^{2} \)
73 \( 1 + (-0.712 + 0.411i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.4 + 6.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (-6.37 + 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.95iT - 97T^{2} \)
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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

−11.52725231137411824688617696446, −10.78043049339239028255670757597, −9.104039437323626111280969704791, −8.673587232546592333485277512197, −7.59612950992408464985536763215, −6.95121915223600220115269340845, −5.75328279374846589751342194051, −4.36053827994510408862840935762, −3.64093055632661606375377933525, −0.72788646199333509684146410544, 1.41341279593547241349394288460, 2.88416286205206929556543583834, 4.24957513527235612455213003342, 5.69211111009483884303674562601, 6.78819525052272579095626987689, 7.64369474420273704237215228252, 8.883396364764229756092567971774, 9.702125117350515567552756505288, 10.98657338298773295963688465066, 11.36109603907913689244579904781

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