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Properties

Label 2-357-119.10-c1-0-15
Degree $2$
Conductor $357$
Sign $0.970 - 0.241i$
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  + (2.11 − 0.278i)2-s + (0.442 + 0.896i)3-s + (2.46 − 0.659i)4-s + (0.750 − 0.657i)5-s + (1.18 + 1.77i)6-s + (0.498 + 2.59i)7-s + (1.08 − 0.448i)8-s + (−0.608 + 0.793i)9-s + (1.40 − 1.59i)10-s + (0.174 − 0.0114i)11-s + (1.68 + 1.91i)12-s + (−2.14 − 2.14i)13-s + (1.77 + 5.35i)14-s + (0.921 + 0.381i)15-s + (−2.25 + 1.30i)16-s + (2.60 − 3.19i)17-s + ⋯
L(s)  = 1  + (1.49 − 0.196i)2-s + (0.255 + 0.517i)3-s + (1.23 − 0.329i)4-s + (0.335 − 0.294i)5-s + (0.483 + 0.723i)6-s + (0.188 + 0.982i)7-s + (0.382 − 0.158i)8-s + (−0.202 + 0.264i)9-s + (0.443 − 0.505i)10-s + (0.0526 − 0.00344i)11-s + (0.485 + 0.553i)12-s + (−0.596 − 0.596i)13-s + (0.475 + 1.43i)14-s + (0.238 + 0.0985i)15-s + (−0.562 + 0.325i)16-s + (0.630 − 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07151 + 0.376623i\)
\(L(\frac12)\) \(\approx\) \(3.07151 + 0.376623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.442 - 0.896i)T \)
7 \( 1 + (-0.498 - 2.59i)T \)
17 \( 1 + (-2.60 + 3.19i)T \)
good2 \( 1 + (-2.11 + 0.278i)T + (1.93 - 0.517i)T^{2} \)
5 \( 1 + (-0.750 + 0.657i)T + (0.652 - 4.95i)T^{2} \)
11 \( 1 + (-0.174 + 0.0114i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (2.14 + 2.14i)T + 13iT^{2} \)
19 \( 1 + (1.07 + 8.17i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (-0.650 - 0.321i)T + (14.0 + 18.2i)T^{2} \)
29 \( 1 + (0.261 + 1.31i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (2.49 - 1.22i)T + (18.8 - 24.5i)T^{2} \)
37 \( 1 + (-0.389 + 5.94i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (0.0270 - 0.135i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-3.28 - 7.93i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (3.14 - 11.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.01 + 3.92i)T + (-13.7 + 51.1i)T^{2} \)
59 \( 1 + (-1.10 - 0.145i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (3.10 + 9.14i)T + (-48.3 + 37.1i)T^{2} \)
67 \( 1 + (1.28 + 0.739i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 - 7.40i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (4.35 + 1.47i)T + (57.9 + 44.4i)T^{2} \)
79 \( 1 + (-1.71 + 3.48i)T + (-48.0 - 62.6i)T^{2} \)
83 \( 1 + (-4.03 + 9.74i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-0.409 - 0.109i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.57 + 1.90i)T + (89.6 - 37.1i)T^{2} \)
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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

−11.59089521363286596377682009788, −11.00703357083509743299530914551, −9.519523577361359604036182697911, −8.996447676423200801269554320109, −7.58306329433715333767347419501, −6.19533585903486808917202320255, −5.18170974402029186765107678646, −4.77752394712166662046106455845, −3.23601220620339545390237754076, −2.40350222658844483917214843353, 1.92798727308181076529837480236, 3.45185637227618825071978151018, 4.27350415570508739821024500928, 5.57847869764126348684822758242, 6.46737951313327845660727080535, 7.28091228812782598656029010014, 8.291621139997980904137138049487, 9.807492418376906637734237557849, 10.64145674476420021628156815276, 11.92207273020319154467101100286

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