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Properties

Label 2-357-119.16-c1-0-15
Degree $2$
Conductor $357$
Sign $0.975 - 0.221i$
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  + (0.0460 − 0.0796i)2-s + (0.866 − 0.5i)3-s + (0.995 + 1.72i)4-s + (2.24 + 1.29i)5-s − 0.0920i·6-s + (0.592 − 2.57i)7-s + 0.367·8-s + (0.499 − 0.866i)9-s + (0.206 − 0.119i)10-s + (−0.491 + 0.283i)11-s + (1.72 + 0.995i)12-s − 4.02·13-s + (−0.178 − 0.165i)14-s + 2.59·15-s + (−1.97 + 3.42i)16-s + (2.53 − 3.25i)17-s + ⋯
L(s)  = 1  + (0.0325 − 0.0563i)2-s + (0.499 − 0.288i)3-s + (0.497 + 0.862i)4-s + (1.00 + 0.580i)5-s − 0.0375i·6-s + (0.224 − 0.974i)7-s + 0.129·8-s + (0.166 − 0.288i)9-s + (0.0653 − 0.0377i)10-s + (−0.148 + 0.0855i)11-s + (0.497 + 0.287i)12-s − 1.11·13-s + (−0.0476 − 0.0443i)14-s + 0.670·15-s + (−0.493 + 0.855i)16-s + (0.614 − 0.789i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.975 - 0.221i$
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: \((2,\ 357,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98338 + 0.222586i\)
\(L(\frac12)\) \(\approx\) \(1.98338 + 0.222586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.592 + 2.57i)T \)
17 \( 1 + (-2.53 + 3.25i)T \)
good2 \( 1 + (-0.0460 + 0.0796i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.24 - 1.29i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.491 - 0.283i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.02T + 13T^{2} \)
19 \( 1 + (2.52 - 4.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.789 - 0.455i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.39iT - 29T^{2} \)
31 \( 1 + (3.87 - 2.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.21 - 0.703i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.36iT - 41T^{2} \)
43 \( 1 - 2.56T + 43T^{2} \)
47 \( 1 + (-2.24 + 3.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.854 - 1.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.95 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.3 + 5.99i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.74 - 8.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (7.71 - 4.45i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.99 + 4.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + (-9.00 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.04iT - 97T^{2} \)
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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

−11.55131233874758299906800670089, −10.39084370987587750204482730401, −9.876663536585438505076132204362, −8.572055680036501779679599039327, −7.38732437901576157751077822431, −7.14839930153259582278966855177, −5.81418747059232355991180205342, −4.22238960294537216458348125811, −2.97730914073072857352780666603, −1.97190255782211056539781376543, 1.75443764182631846248132821287, 2.68967731232294028455710463288, 4.78991181940963391056458207725, 5.47567024452449329384864289900, 6.38913196636998789500108358521, 7.72212236094475081814989455050, 8.979434637651629381613310741751, 9.487017342888074045584339469852, 10.33787093516307479297562353345, 11.27845011044004174715657144636

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