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Properties

Label 2-357-119.67-c1-0-3
Degree $2$
Conductor $357$
Sign $-0.828 + 0.559i$
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  + (1.37 + 2.38i)2-s + (−0.866 − 0.5i)3-s + (−2.78 + 4.82i)4-s + (−0.416 + 0.240i)5-s − 2.75i·6-s + (−2.60 + 0.460i)7-s − 9.83·8-s + (0.499 + 0.866i)9-s + (−1.14 − 0.661i)10-s + (2.87 + 1.65i)11-s + (4.82 − 2.78i)12-s − 0.614·13-s + (−4.68 − 5.57i)14-s + 0.481·15-s + (−7.95 − 13.7i)16-s + (0.167 − 4.11i)17-s + ⋯
L(s)  = 1  + (0.972 + 1.68i)2-s + (−0.499 − 0.288i)3-s + (−1.39 + 2.41i)4-s + (−0.186 + 0.107i)5-s − 1.12i·6-s + (−0.984 + 0.173i)7-s − 3.47·8-s + (0.166 + 0.288i)9-s + (−0.362 − 0.209i)10-s + (0.865 + 0.499i)11-s + (1.39 − 0.804i)12-s − 0.170·13-s + (−1.25 − 1.49i)14-s + 0.124·15-s + (−1.98 − 3.44i)16-s + (0.0406 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.828 + 0.559i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.828 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.346503 - 1.13151i\)
\(L(\frac12)\) \(\approx\) \(0.346503 - 1.13151i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 0.460i)T \)
17 \( 1 + (-0.167 + 4.11i)T \)
good2 \( 1 + (-1.37 - 2.38i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.416 - 0.240i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.87 - 1.65i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.614T + 13T^{2} \)
19 \( 1 + (-1.77 - 3.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.94 - 3.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.80iT - 29T^{2} \)
31 \( 1 + (-6.22 - 3.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.86 + 2.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.388iT - 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 + (-4.54 - 7.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.365 + 0.633i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.79 - 3.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.18 + 4.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.97 + 3.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.11iT - 71T^{2} \)
73 \( 1 + (13.8 + 8.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.0 + 6.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.63T + 83T^{2} \)
89 \( 1 + (8.37 + 14.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.51iT - 97T^{2} \)
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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

−12.22832413901003103477201134997, −11.71615534885742382763958031587, −9.811498303433137499023501626151, −9.001205144594486344484510186472, −7.67754829325636945828010643117, −7.08029364282599512690200830494, −6.22713618874719346772423181858, −5.47335709666117716027321055429, −4.29319719325675643624585488609, −3.23624337578634497052167513074, 0.63898557829987128892917796907, 2.50541146359108699972721539231, 3.84783980554889653396242873810, 4.33699588055777833413588594878, 5.84477089138643181078621193693, 6.38610588670798759603000676316, 8.557731069359773180312609975586, 9.772433808636645591664862013409, 10.07491966874126641454099448221, 11.14827434929000889154685260659

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