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Properties

Label 2-350-35.9-c3-0-19
Degree $2$
Conductor $350$
Sign $0.895 - 0.445i$
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  + (1.73 + i)2-s + (−7.22 + 4.16i)3-s + (1.99 + 3.46i)4-s − 16.6·6-s + (−8.19 − 16.6i)7-s + 7.99i·8-s + (21.2 − 36.8i)9-s + (13.1 + 22.7i)11-s + (−28.8 − 16.6i)12-s − 38.1i·13-s + (2.41 − 36.9i)14-s + (−8 + 13.8i)16-s + (−2.07 + 1.19i)17-s + (73.7 − 42.5i)18-s + (49.8 − 86.4i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.38 + 0.802i)3-s + (0.249 + 0.433i)4-s − 1.13·6-s + (−0.442 − 0.896i)7-s + 0.353i·8-s + (0.788 − 1.36i)9-s + (0.360 + 0.623i)11-s + (−0.694 − 0.401i)12-s − 0.814i·13-s + (0.0460 − 0.705i)14-s + (−0.125 + 0.216i)16-s + (−0.0295 + 0.0170i)17-s + (0.965 − 0.557i)18-s + (0.602 − 1.04i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.895 - 0.445i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.449295238\)
\(L(\frac12)\) \(\approx\) \(1.449295238\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
5 \( 1 \)
7 \( 1 + (8.19 + 16.6i)T \)
good3 \( 1 + (7.22 - 4.16i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (-13.1 - 22.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 38.1iT - 2.19e3T^{2} \)
17 \( 1 + (2.07 - 1.19i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-49.8 + 86.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (88.1 + 50.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 223.T + 2.43e4T^{2} \)
31 \( 1 + (-133. - 231. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-314. - 181. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 94.9T + 6.89e4T^{2} \)
43 \( 1 - 406. iT - 7.95e4T^{2} \)
47 \( 1 + (296. + 171. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-470. + 271. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (196. + 340. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-252. + 436. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (47.5 - 27.4i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 889.T + 3.57e5T^{2} \)
73 \( 1 + (-442. + 255. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-136. + 237. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 525. iT - 5.71e5T^{2} \)
89 \( 1 + (-827. + 1.43e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 5.76iT - 9.12e5T^{2} \)
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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

−11.19440015579772260135841623414, −10.26104082434845365939222193427, −9.741967577675620396728539448984, −8.115528805464494663286723168655, −6.80222498646212352893543002508, −6.30206850630133072112846070885, −4.97718163711057130548837737050, −4.49427638231673156013827695084, −3.20988517213239214754403609966, −0.68106415984842921853941146169, 0.943043875559843830092030799257, 2.32389694145188210635239070588, 3.97447866269778886346209480839, 5.36737162450854691911827635646, 6.04847121292086856432766280035, 6.65105670804791343327458021546, 7.963314067232480887655545044799, 9.358776556749998640566611563873, 10.36441129227080838208593454730, 11.53371939497772903202031098779

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