Properties

Label 2-350-35.4-c1-0-3
Degree $2$
Conductor $350$
Sign $0.982 - 0.185i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (2.59 − 0.5i)7-s + 0.999i·8-s + (−1 − 1.73i)9-s + (3 − 5.19i)11-s + (0.866 − 0.499i)12-s + 4i·13-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (1.73 + i)18-s + (1 + 1.73i)19-s + (2.5 + 0.866i)21-s + 6i·22-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (0.981 − 0.188i)7-s + 0.353i·8-s + (−0.333 − 0.577i)9-s + (0.904 − 1.56i)11-s + (0.249 − 0.144i)12-s + 1.10i·13-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.408 + 0.235i)18-s + (0.229 + 0.397i)19-s + (0.545 + 0.188i)21-s + 1.27i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30323 + 0.122012i\)
\(L(\frac12)\) \(\approx\) \(1.30323 + 0.122012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 + 0.5i)T \)
good3 \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (13.8 + 8i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−11.37675653877731911265528575407, −10.60856472007741682197953321912, −9.232695041344867727798746189517, −8.848983659480269572644162200862, −8.022002325648279848114577201906, −6.77879563882621567603608728916, −5.87695300798058890414788620154, −4.44554137346471178621025554449, −3.19722996698759305301979027557, −1.32488356409602667029950332488, 1.60410686320432044895721850408, 2.68958597263851523776031445531, 4.31567447534530684520860399671, 5.51152632267334156610184901483, 7.19974142641896489906865942100, 7.71978488539889094969323202128, 8.699408839989360034408123144758, 9.493775503606488602488951141454, 10.57183893618466891297101052028, 11.36059193349665317423576058521

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