↑

Properties

Label 2-357-7.4-c1-0-21
Degree $2$
Conductor $357$
Sign $0.0918 - 0.995i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.19 − 2.07i)2-s + (0.5 − 0.866i)3-s + (−1.87 + 3.23i)4-s + (−2.18 − 3.77i)5-s − 2.39·6-s + (−2.64 − 0.0756i)7-s + 4.17·8-s + (−0.499 − 0.866i)9-s + (−5.22 + 9.05i)10-s + (1.09 − 1.89i)11-s + (1.87 + 3.23i)12-s + 4.97·13-s + (3.01 + 5.57i)14-s − 4.36·15-s + (−1.25 − 2.17i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.847 − 1.46i)2-s + (0.288 − 0.499i)3-s + (−0.935 + 1.61i)4-s + (−0.975 − 1.69i)5-s − 0.978·6-s + (−0.999 − 0.0285i)7-s + 1.47·8-s + (−0.166 − 0.288i)9-s + (−1.65 + 2.86i)10-s + (0.330 − 0.572i)11-s + (0.539 + 0.935i)12-s + 1.38·13-s + (0.804 + 1.49i)14-s − 1.12·15-s + (−0.314 − 0.544i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0918 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0918 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.0918 - 0.995i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.0918 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319375 + 0.291273i\)
\(L(\frac12)\) \(\approx\) \(0.319375 + 0.291273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0756i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.19 + 2.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.18 + 3.77i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.09 + 1.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
19 \( 1 + (1.28 + 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.98 - 5.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.59T + 29T^{2} \)
31 \( 1 + (-1.62 + 2.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.41 + 5.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.32T + 41T^{2} \)
43 \( 1 - 3.15T + 43T^{2} \)
47 \( 1 + (0.0918 + 0.159i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.42 + 2.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.78 + 6.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.801 - 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + (4.18 - 7.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.55 + 6.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.625T + 83T^{2} \)
89 \( 1 + (0.479 + 0.830i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.43T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.06576181783585143848178777321, −9.564115289410223960179260913174, −8.915624442122579103478747035645, −8.511936359509815135432204445348, −7.43550303872562175998574926096, −5.77899321804929486339871563143, −3.98792669886008255647835797152, −3.40893005319167974646659878318, −1.52427912430668154755287805064, −0.40018426356627841106979918160, 3.08884026888205276998788052551, 4.11887457250967327347275176072, 5.96208403669599600191709899582, 6.70295998246031740298156125683, 7.30775638693305682816693468816, 8.333247610047255195785601296591, 9.161681317993716209635801735334, 10.23552232679570028812659826735, 10.72139462605172028859468517813, 11.91896394766051171201352798286

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