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Properties

Label 2-350-35.17-c1-0-4
Degree $2$
Conductor $350$
Sign $0.823 - 0.567i$
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 + (−1.95 + 0.523i)3-s + (−0.866 − 0.499i)4-s − 2.02i·6-s + (1.90 − 1.83i)7-s + (0.707 − 0.707i)8-s + (0.941 − 0.543i)9-s + (2.01 − 3.49i)11-s + (1.95 + 0.523i)12-s + (0.204 + 0.204i)13-s + (1.28 + 2.31i)14-s + (0.500 + 0.866i)16-s + (0.527 + 1.97i)17-s + (0.281 + 1.05i)18-s + (3.10 + 5.37i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−1.12 + 0.302i)3-s + (−0.433 − 0.249i)4-s − 0.825i·6-s + (0.718 − 0.695i)7-s + (0.249 − 0.249i)8-s + (0.313 − 0.181i)9-s + (0.609 − 1.05i)11-s + (0.563 + 0.151i)12-s + (0.0568 + 0.0568i)13-s + (0.343 + 0.618i)14-s + (0.125 + 0.216i)16-s + (0.128 + 0.477i)17-s + (0.0663 + 0.247i)18-s + (0.711 + 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.823 - 0.567i$
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: \((2,\ 350,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840961 + 0.261897i\)
\(L(\frac12)\) \(\approx\) \(0.840961 + 0.261897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 1.83i)T \)
good3 \( 1 + (1.95 - 0.523i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.01 + 3.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.204 - 0.204i)T + 13iT^{2} \)
17 \( 1 + (-0.527 - 1.97i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.10 - 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.38 - 1.17i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 7.15iT - 29T^{2} \)
31 \( 1 + (-6.33 - 3.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.19 - 4.46i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \)
47 \( 1 + (-0.304 - 0.0815i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.14 + 8.00i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.427 - 0.740i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.99 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.05 - 0.817i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + (-11.1 + 2.98i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.39 + 2.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.85 + 3.85i)T + 83iT^{2} \)
89 \( 1 + (-1.53 - 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.63 - 6.63i)T - 97iT^{2} \)
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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

−11.44674968993254398521730368442, −10.68955019931031684570314069021, −9.900245515846230291742901879009, −8.589246911881118635037232073221, −7.80126720469220488954684020343, −6.57357530097554951963368218245, −5.78833842187385256426017315324, −4.88918744984623660274355700347, −3.74032964758781603968502054714, −1.01456766937658253366744220009, 1.17202582796666038755636928122, 2.74412981004959317324646145688, 4.61556477889089899136805088059, 5.27751001509911802990710429717, 6.57196167164494173693037244755, 7.53295677397267312877559193175, 8.892706012329294469169067485795, 9.565393129314472439582726310263, 10.92561552801037501952999100667, 11.33504150889766330509570069429

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