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Properties

Label 2-357-7.2-c1-0-20
Degree $2$
Conductor $357$
Sign $-0.994 + 0.102i$
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  + (1.10 − 1.91i)2-s + (−0.5 − 0.866i)3-s + (−1.43 − 2.49i)4-s + (−0.137 + 0.238i)5-s − 2.20·6-s + (−0.103 − 2.64i)7-s − 1.93·8-s + (−0.499 + 0.866i)9-s + (0.304 + 0.526i)10-s + (−1.10 − 1.91i)11-s + (−1.43 + 2.49i)12-s − 1.80·13-s + (−5.17 − 2.71i)14-s + 0.275·15-s + (0.741 − 1.28i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.780 − 1.35i)2-s + (−0.288 − 0.499i)3-s + (−0.718 − 1.24i)4-s + (−0.0615 + 0.106i)5-s − 0.901·6-s + (−0.0393 − 0.999i)7-s − 0.683·8-s + (−0.166 + 0.288i)9-s + (0.0961 + 0.166i)10-s + (−0.332 − 0.576i)11-s + (−0.415 + 0.718i)12-s − 0.501·13-s + (−1.38 − 0.726i)14-s + 0.0711·15-s + (0.185 − 0.321i)16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.994 + 0.102i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0845492 - 1.64519i\)
\(L(\frac12)\) \(\approx\) \(0.0845492 - 1.64519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.103 + 2.64i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.10 + 1.91i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.137 - 0.238i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.10 + 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.80T + 13T^{2} \)
19 \( 1 + (2.07 - 3.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.966 + 1.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 + (1.40 + 2.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.70 + 4.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 2.84T + 43T^{2} \)
47 \( 1 + (-2.17 + 3.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.03 + 1.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.49 - 4.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.27 - 7.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.72 + 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.95T + 71T^{2} \)
73 \( 1 + (-3.68 - 6.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.22 + 2.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.89T + 83T^{2} \)
89 \( 1 + (7.92 - 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.86T + 97T^{2} \)
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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

−10.92102793150893043259920508686, −10.66173699167857925804437320241, −9.623626858087075950086446467658, −8.124745750765802271447953756442, −7.14750828261365387063634156387, −5.88954492499370352686614337815, −4.71888797479329130246682177320, −3.70767756047626342755291259463, −2.51042211698991662965641108304, −0.979365056853101942037530076162, 2.79047753852486087333228231006, 4.50554489988047718856619859156, 5.00539478066781348784163397624, 6.05464591202266850082157460203, 6.87023914102365961042881497010, 7.994437279525268468480837251976, 8.886985415955559216449078372742, 9.919402970940691920423899867574, 11.06677429132986786648931472334, 12.28447277617557027175137514428

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