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Properties

Label 2-357-119.100-c1-0-16
Degree $2$
Conductor $357$
Sign $-0.197 + 0.980i$
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  + (−1.68 + 0.452i)2-s + (0.608 − 0.793i)3-s + (0.909 − 0.525i)4-s + (0.277 − 2.10i)5-s + (−0.668 + 1.61i)6-s + (2.14 − 1.55i)7-s + (1.17 − 1.17i)8-s + (−0.258 − 0.965i)9-s + (0.484 + 3.67i)10-s + (−5.06 + 0.667i)11-s + (0.137 − 1.04i)12-s + 0.853i·13-s + (−2.91 + 3.58i)14-s + (−1.50 − 1.50i)15-s + (−2.49 + 4.32i)16-s + (2.60 − 3.19i)17-s + ⋯
L(s)  = 1  + (−1.19 + 0.319i)2-s + (0.351 − 0.458i)3-s + (0.454 − 0.262i)4-s + (0.123 − 0.941i)5-s + (−0.272 + 0.658i)6-s + (0.809 − 0.586i)7-s + (0.414 − 0.414i)8-s + (−0.0862 − 0.321i)9-s + (0.153 + 1.16i)10-s + (−1.52 + 0.201i)11-s + (0.0395 − 0.300i)12-s + 0.236i·13-s + (−0.778 + 0.958i)14-s + (−0.387 − 0.387i)15-s + (−0.624 + 1.08i)16-s + (0.630 − 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431379 - 0.526972i\)
\(L(\frac12)\) \(\approx\) \(0.431379 - 0.526972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (-2.14 + 1.55i)T \)
17 \( 1 + (-2.60 + 3.19i)T \)
good2 \( 1 + (1.68 - 0.452i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.277 + 2.10i)T + (-4.82 - 1.29i)T^{2} \)
11 \( 1 + (5.06 - 0.667i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 - 0.853iT - 13T^{2} \)
19 \( 1 + (2.98 - 0.799i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.68 + 4.80i)T + (-5.95 + 22.2i)T^{2} \)
29 \( 1 + (3.11 - 1.28i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-4.55 + 5.93i)T + (-8.02 - 29.9i)T^{2} \)
37 \( 1 + (3.99 + 0.525i)T + (35.7 + 9.57i)T^{2} \)
41 \( 1 + (-2.06 - 0.854i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4.32 + 4.32i)T - 43iT^{2} \)
47 \( 1 + (1.22 + 0.708i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.02 + 7.56i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.92 + 1.58i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.98 - 1.51i)T + (15.7 - 58.9i)T^{2} \)
67 \( 1 + (-3.97 - 6.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.52 - 3.67i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.66 - 3.57i)T + (18.8 + 70.5i)T^{2} \)
79 \( 1 + (1.69 + 2.21i)T + (-20.4 + 76.3i)T^{2} \)
83 \( 1 + (-8.40 - 8.40i)T + 83iT^{2} \)
89 \( 1 + (-13.8 - 7.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.6 - 4.84i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.84516973883784771290363625085, −10.09464510887088399231648653699, −9.151501886380842775616618351247, −8.143562327911319282566767420281, −7.920793522365414977910334421144, −6.86882895500092934385459036286, −5.30035602687863789973074497931, −4.24590443907789125141206997088, −2.13025898771611903879656623876, −0.66213358550955594070765719959, 1.98832410537360874708159081510, 3.04509886347185591410945632120, 4.82044691804603167046664193649, 5.87921463139907014284916359482, 7.59083956672449737537522366361, 8.080508465808645665793147477678, 8.948675029623045491383456798331, 10.04488628287750063300224522017, 10.62159031973184265704781732571, 11.11158114291744384973043020445

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