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Properties

Label 2-350-35.27-c3-0-24
Degree $2$
Conductor $350$
Sign $-0.998 - 0.0463i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 1.41i)2-s + (−5.81 − 5.81i)3-s + 4.00i·4-s + 16.4i·6-s + (17.8 + 5.08i)7-s + (5.65 − 5.65i)8-s + 40.5i·9-s − 13.1·11-s + (23.2 − 23.2i)12-s + (−16.3 − 16.3i)13-s + (−17.9 − 32.3i)14-s − 16.0·16-s + (76.6 − 76.6i)17-s + (57.3 − 57.3i)18-s + 29.7·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.11 − 1.11i)3-s + 0.500i·4-s + 1.11i·6-s + (0.961 + 0.274i)7-s + (0.250 − 0.250i)8-s + 1.50i·9-s − 0.361·11-s + (0.559 − 0.559i)12-s + (−0.348 − 0.348i)13-s + (−0.343 − 0.618i)14-s − 0.250·16-s + (1.09 − 1.09i)17-s + (0.751 − 0.751i)18-s + 0.359·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0463i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.998 - 0.0463i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.998 - 0.0463i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7272694429\)
\(L(\frac12)\) \(\approx\) \(0.7272694429\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 1.41i)T \)
5 \( 1 \)
7 \( 1 + (-17.8 - 5.08i)T \)
good3 \( 1 + (5.81 + 5.81i)T + 27iT^{2} \)
11 \( 1 + 13.1T + 1.33e3T^{2} \)
13 \( 1 + (16.3 + 16.3i)T + 2.19e3iT^{2} \)
17 \( 1 + (-76.6 + 76.6i)T - 4.91e3iT^{2} \)
19 \( 1 - 29.7T + 6.85e3T^{2} \)
23 \( 1 + (-134. + 134. i)T - 1.21e4iT^{2} \)
29 \( 1 - 136. iT - 2.43e4T^{2} \)
31 \( 1 + 176. iT - 2.97e4T^{2} \)
37 \( 1 + (283. + 283. i)T + 5.06e4iT^{2} \)
41 \( 1 + 48.3iT - 6.89e4T^{2} \)
43 \( 1 + (136. - 136. i)T - 7.95e4iT^{2} \)
47 \( 1 + (373. - 373. i)T - 1.03e5iT^{2} \)
53 \( 1 + (113. - 113. i)T - 1.48e5iT^{2} \)
59 \( 1 - 642.T + 2.05e5T^{2} \)
61 \( 1 + 254. iT - 2.26e5T^{2} \)
67 \( 1 + (-87.5 - 87.5i)T + 3.00e5iT^{2} \)
71 \( 1 + 81.2T + 3.57e5T^{2} \)
73 \( 1 + (505. + 505. i)T + 3.89e5iT^{2} \)
79 \( 1 - 873. iT - 4.93e5T^{2} \)
83 \( 1 + (260. + 260. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.28e3T + 7.04e5T^{2} \)
97 \( 1 + (-721. + 721. i)T - 9.12e5iT^{2} \)
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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

−10.98664403292631339300674407508, −9.857457094623705041059036598472, −8.587590578727693645094743314911, −7.60652974576297628954688237044, −7.01418682958329611631918461520, −5.60459903349860025886449615071, −4.88038988989172682038125495728, −2.80009894591222871805106346935, −1.46704060641875737895970590835, −0.40314014843422551424482918779, 1.32643252260920359851070654947, 3.69601084572150674409124387219, 5.04428635093287679749428280971, 5.36212455020855657330091789471, 6.68110857524965099950846920280, 7.78497722727617778425476791944, 8.790947511768471433421265637840, 10.03502574959158448749570560987, 10.34024376144379620580152225271, 11.40150365167378014451635005102

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