↑

Properties

Label 2-350-35.13-c1-0-3
Degree $2$
Conductor $350$
Sign $0.757 - 0.652i$
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 + (0.541 − 0.541i)3-s − 1.00i·4-s + 0.765i·6-s + (2.14 + 1.55i)7-s + (0.707 + 0.707i)8-s + 2.41i·9-s − 2.82·11-s + (−0.541 − 0.541i)12-s + (2.83 − 2.83i)13-s + (−2.61 + 0.414i)14-s − 1.00·16-s + (1.53 + 1.53i)17-s + (−1.70 − 1.70i)18-s + 7.07·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.312 − 0.312i)3-s − 0.500i·4-s + 0.312i·6-s + (0.809 + 0.587i)7-s + (0.250 + 0.250i)8-s + 0.804i·9-s − 0.852·11-s + (−0.156 − 0.156i)12-s + (0.786 − 0.786i)13-s + (−0.698 + 0.110i)14-s − 0.250·16-s + (0.371 + 0.371i)17-s + (−0.402 − 0.402i)18-s + 1.62·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.757 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17437 + 0.435866i\)
\(L(\frac12)\) \(\approx\) \(1.17437 + 0.435866i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 - 1.55i)T \)
good3 \( 1 + (-0.541 + 0.541i)T - 3iT^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (-2.83 + 2.83i)T - 13iT^{2} \)
17 \( 1 + (-1.53 - 1.53i)T + 17iT^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 + (-2.41 - 2.41i)T + 23iT^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 - 3.69iT - 31T^{2} \)
37 \( 1 + (5.41 - 5.41i)T - 37iT^{2} \)
41 \( 1 + 1.53iT - 41T^{2} \)
43 \( 1 + (4 + 4i)T + 43iT^{2} \)
47 \( 1 + (2.61 + 2.61i)T + 47iT^{2} \)
53 \( 1 + (0.242 + 0.242i)T + 53iT^{2} \)
59 \( 1 + 3.82T + 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
67 \( 1 + (-6.48 + 6.48i)T - 67iT^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + (-4.77 + 4.77i)T - 73iT^{2} \)
79 \( 1 - 9.07iT - 79T^{2} \)
83 \( 1 + (5.45 - 5.45i)T - 83iT^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 + (11.0 + 11.0i)T + 97iT^{2} \)
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

−11.38977339064352297700326525093, −10.63015065569898900094403542630, −9.665657446100879057664869751964, −8.336392225204929539410008376054, −8.097146017103375501992944320506, −7.12067485623069608369218744616, −5.58963155649491886803430162202, −5.09097525740223001514356292096, −3.06382179860592717388841713459, −1.57139613170821201271063506068, 1.22389050796183656808670881518, 2.99232099952717594657840266269, 4.06508666829521439326592582930, 5.28779725415840877113771699576, 6.87163135705052328971138646938, 7.78587091444472885000816954458, 8.749898680838176927216734855615, 9.542332314869434815547195397423, 10.45064210210823842356091886714, 11.30120102424052541061098922693

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