↑

Properties

Label 2-350-35.27-c1-0-9
Degree $2$
Conductor $350$
Sign $0.640 + 0.768i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (0.707 + 0.707i)2-s + (−1.32 − 1.32i)3-s + 1.00i·4-s − 1.88i·6-s + (1.36 − 2.26i)7-s + (−0.707 + 0.707i)8-s + 0.535i·9-s + 1.73·11-s + (1.32 − 1.32i)12-s + (−3.63 − 3.63i)13-s + (2.56 − 0.633i)14-s − 1.00·16-s + (2.30 − 2.30i)17-s + (−0.378 + 0.378i)18-s + 3.25·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.767 − 0.767i)3-s + 0.500i·4-s − 0.767i·6-s + (0.517 − 0.855i)7-s + (−0.250 + 0.250i)8-s + 0.178i·9-s + 0.522·11-s + (0.383 − 0.383i)12-s + (−1.00 − 1.00i)13-s + (0.686 − 0.169i)14-s − 0.250·16-s + (0.558 − 0.558i)17-s + (−0.0893 + 0.0893i)18-s + 0.747·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.640 + 0.768i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.640 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22027 - 0.571448i\)
\(L(\frac12)\) \(\approx\) \(1.22027 - 0.571448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-1.36 + 2.26i)T \)
good3 \( 1 + (1.32 + 1.32i)T + 3iT^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + (3.63 + 3.63i)T + 13iT^{2} \)
17 \( 1 + (-2.30 + 2.30i)T - 17iT^{2} \)
19 \( 1 - 3.25T + 19T^{2} \)
23 \( 1 + (-5.79 + 5.79i)T - 23iT^{2} \)
29 \( 1 + 4.73iT - 29T^{2} \)
31 \( 1 - 8.89iT - 31T^{2} \)
37 \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \)
41 \( 1 + 5.64iT - 41T^{2} \)
43 \( 1 + (2.44 - 2.44i)T - 43iT^{2} \)
47 \( 1 + (6.29 - 6.29i)T - 47iT^{2} \)
53 \( 1 + (10.0 - 10.0i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6.51iT - 61T^{2} \)
67 \( 1 + (-7.02 - 7.02i)T + 67iT^{2} \)
71 \( 1 - 8.19T + 71T^{2} \)
73 \( 1 + (-7.62 - 7.62i)T + 73iT^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (-3.98 - 3.98i)T + 83iT^{2} \)
89 \( 1 + 5.64T + 89T^{2} \)
97 \( 1 + (-7.26 + 7.26i)T - 97iT^{2} \)
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

−11.59448378494578098140460388792, −10.68213633943918633126720812683, −9.542447946792740919180206690429, −8.124859934780388011629344663751, −7.26812732031327559785906219580, −6.68594276460573538431174773141, −5.46745990138002377353468420054, −4.65484154818295144124209241475, −3.10145295318276438986256442731, −0.951048946975474971975529853595, 1.90562284567538727278292147327, 3.54396450783336187426042994721, 4.86312964463564479848111931220, 5.29749715604075918803064947048, 6.48416329169891465763998205976, 7.85464531111582443357213437794, 9.358437979042100946432487935472, 9.735007692914236823307477998848, 11.07343388473527943239338116266, 11.51215708010502603439019002600

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