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Properties

Label 2-350-35.17-c1-0-1
Degree $2$
Conductor $350$
Sign $0.419 - 0.907i$
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 + (−1.13 + 0.304i)3-s + (−0.866 − 0.499i)4-s + 1.17i·6-s + (−0.698 + 2.55i)7-s + (−0.707 + 0.707i)8-s + (−1.40 + 0.810i)9-s + (−0.371 + 0.643i)11-s + (1.13 + 0.304i)12-s + (2.05 + 2.05i)13-s + (2.28 + 1.33i)14-s + (0.500 + 0.866i)16-s + (1.69 + 6.33i)17-s + (0.419 + 1.56i)18-s + (−0.946 − 1.63i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.655 + 0.175i)3-s + (−0.433 − 0.249i)4-s + 0.479i·6-s + (−0.264 + 0.964i)7-s + (−0.249 + 0.249i)8-s + (−0.467 + 0.270i)9-s + (−0.112 + 0.194i)11-s + (0.327 + 0.0877i)12-s + (0.570 + 0.570i)13-s + (0.610 + 0.356i)14-s + (0.125 + 0.216i)16-s + (0.411 + 1.53i)17-s + (0.0988 + 0.368i)18-s + (−0.217 − 0.375i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.419 - 0.907i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.419 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656069 + 0.419790i\)
\(L(\frac12)\) \(\approx\) \(0.656069 + 0.419790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.698 - 2.55i)T \)
good3 \( 1 + (1.13 - 0.304i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.371 - 0.643i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \)
17 \( 1 + (-1.69 - 6.33i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.946 + 1.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.11 + 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 9.69iT - 29T^{2} \)
31 \( 1 + (-2.96 - 1.71i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.691 + 2.58i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.817iT - 41T^{2} \)
43 \( 1 + (1.59 - 1.59i)T - 43iT^{2} \)
47 \( 1 + (4.54 + 1.21i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.29 + 4.81i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.27 + 2.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.25 + 3.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.2 - 3.54i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + (-8.54 + 2.29i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.70 + 3.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.23 - 9.23i)T + 83iT^{2} \)
89 \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \)
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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

−11.67145649078981581458741093690, −10.82112677566075630753487380707, −10.10851092623757838713441012007, −8.910187302377500907646292640611, −8.247694030407528488316657907370, −6.45647138635391451597995084507, −5.73178325174526342628729018418, −4.70321340966733271364425973192, −3.35949783084190610676858461640, −1.92904891740329212391392527082, 0.54910380064854861541863576838, 3.19205720911528063163332342319, 4.43586003585580542737566304042, 5.69401930744175028640768536617, 6.34256904996187912563372995392, 7.43980315071607289214372422865, 8.224296514388466645977298455935, 9.521577482092347384539195613214, 10.36888428013965575467507020682, 11.49076425570116698344692599822

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