Properties

Label 2-350-35.24-c2-0-17
Degree $2$
Conductor $350$
Sign $-0.859 + 0.510i$
Analytic cond. $9.53680$
Root an. cond. $3.08817$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (−0.358 + 0.621i)3-s + (0.999 − 1.73i)4-s − 1.01i·6-s + (−3.16 − 6.24i)7-s + 2.82i·8-s + (4.24 + 7.34i)9-s + (2.37 − 4.11i)11-s + (0.717 + 1.24i)12-s − 15.2·13-s + (8.29 + 5.40i)14-s + (−2.00 − 3.46i)16-s + (−1.88 + 3.25i)17-s + (−10.3 − 6i)18-s + (−3.62 + 2.09i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.119 + 0.207i)3-s + (0.249 − 0.433i)4-s − 0.169i·6-s + (−0.452 − 0.891i)7-s + 0.353i·8-s + (0.471 + 0.816i)9-s + (0.216 − 0.374i)11-s + (0.0597 + 0.103i)12-s − 1.17·13-s + (0.592 + 0.386i)14-s + (−0.125 − 0.216i)16-s + (−0.110 + 0.191i)17-s + (−0.577 − 0.333i)18-s + (−0.190 + 0.110i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.859 + 0.510i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.859 + 0.510i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0214066 - 0.0779120i\)
\(L(\frac12)\) \(\approx\) \(0.0214066 - 0.0779120i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (3.16 + 6.24i)T \)
good3 \( 1 + (0.358 - 0.621i)T + (-4.5 - 7.79i)T^{2} \)
11 \( 1 + (-2.37 + 4.11i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 15.2T + 169T^{2} \)
17 \( 1 + (1.88 - 3.25i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (3.62 - 2.09i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (24.0 - 13.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 3.51T + 841T^{2} \)
31 \( 1 + (42.3 + 24.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-2.54 + 1.47i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 27.9iT - 1.68e3T^{2} \)
43 \( 1 + 10.4iT - 1.84e3T^{2} \)
47 \( 1 + (26.3 + 45.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (48.4 + 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (33.5 + 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (78.3 - 45.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.9 + 17.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 36.4T + 5.04e3T^{2} \)
73 \( 1 + (26.3 - 45.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.8 + 29.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 127.T + 6.88e3T^{2} \)
89 \( 1 + (-43.5 + 25.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 101.T + 9.40e3T^{2} \)
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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

−10.59503001053447174008796371035, −10.00776639354598340775682008186, −9.175341127472365898743049547342, −7.80971068250043801347395540866, −7.32608686411557314689308051156, −6.16154169418037837857115746272, −4.97063117932728311880090463562, −3.76859034836191955535439119905, −1.95039519993410911285418188148, −0.04315617580185702000233710722, 1.86615657410001941036490504594, 3.13994827685516196717009717283, 4.59737095124910731651182159925, 6.07377553370953479207128605812, 6.94456582297181891869139258808, 7.945442561514337156451173684536, 9.241304285372021682058268966534, 9.551036264264720054602098891652, 10.63033961811036099331069519488, 11.84407224801582562106491043067

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