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Properties

Label 2-350-35.33-c1-0-6
Degree $2$
Conductor $350$
Sign $0.997 + 0.0672i$
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  + (−0.258 − 0.965i)2-s + (2.63 + 0.707i)3-s + (−0.866 + 0.499i)4-s − 2.73i·6-s + (−1.48 + 2.19i)7-s + (0.707 + 0.707i)8-s + (3.86 + 2.23i)9-s + (2.36 + 4.09i)11-s + (−2.63 + 0.707i)12-s + (2.96 − 2.96i)13-s + (2.49 + 0.866i)14-s + (0.500 − 0.866i)16-s + (1.67 − 6.24i)17-s + (1.15 − 4.31i)18-s + (−5.46 + 4.73i)21-s + (3.34 − 3.34i)22-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (1.52 + 0.408i)3-s + (−0.433 + 0.249i)4-s − 1.11i·6-s + (−0.560 + 0.827i)7-s + (0.249 + 0.249i)8-s + (1.28 + 0.744i)9-s + (0.713 + 1.23i)11-s + (−0.761 + 0.204i)12-s + (0.822 − 0.822i)13-s + (0.668 + 0.231i)14-s + (0.125 − 0.216i)16-s + (0.405 − 1.51i)17-s + (0.272 − 1.01i)18-s + (−1.19 + 1.03i)21-s + (0.713 − 0.713i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.997 + 0.0672i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.997 + 0.0672i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85974 - 0.0625847i\)
\(L(\frac12)\) \(\approx\) \(1.85974 - 0.0625847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.48 - 2.19i)T \)
good3 \( 1 + (-2.63 - 0.707i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.96 + 2.96i)T - 13iT^{2} \)
17 \( 1 + (-1.67 + 6.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.01 - 1.34i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.79 + 6.69i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.803iT - 41T^{2} \)
43 \( 1 + (5.13 + 5.13i)T + 43iT^{2} \)
47 \( 1 + (9.58 - 2.56i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.568 + 2.12i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.90 + 3.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.29 - 5.36i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.7 - 3.67i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 + (-7.72 - 2.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.13 + 4.69i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 + (-0.401 + 0.696i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.64 - 6.64i)T + 97iT^{2} \)
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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

−11.54418910840333561533307309719, −10.12353515327088748810546360433, −9.599284107041178644545249781481, −8.929779393493783836767350979001, −8.091976468288633274620234708570, −6.99351369660970661912046025228, −5.30715693898836143034401244208, −3.90388247441795699585783526676, −3.07540881957292288588474285259, −1.99146174291851267572915043097, 1.50302633749061985291086479363, 3.44056143014029868159285001608, 4.02623185567129203200870764390, 6.15847159412479448098707401686, 6.72359331798955613644164665923, 8.091156270991594211129126570502, 8.364214630655857241065422892706, 9.373863800583300320961064040743, 10.21124818775309672146883618323, 11.45737338949801074728561298854

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