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Properties

Label 2-350-175.83-c1-0-7
Degree $2$
Conductor $350$
Sign $0.891 + 0.453i$
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  + (−0.891 − 0.453i)2-s + (0.115 − 0.728i)3-s + (0.587 + 0.809i)4-s + (2.05 − 0.881i)5-s + (−0.433 + 0.596i)6-s + (0.185 + 2.63i)7-s + (−0.156 − 0.987i)8-s + (2.33 + 0.759i)9-s + (−2.23 − 0.147i)10-s + (−0.985 − 3.03i)11-s + (0.656 − 0.334i)12-s + (2.18 + 4.28i)13-s + (1.03 − 2.43i)14-s + (−0.404 − 1.59i)15-s + (−0.309 + 0.951i)16-s + (0.642 + 4.05i)17-s + ⋯
L(s)  = 1  + (−0.630 − 0.321i)2-s + (0.0665 − 0.420i)3-s + (0.293 + 0.404i)4-s + (0.919 − 0.394i)5-s + (−0.176 + 0.243i)6-s + (0.0702 + 0.997i)7-s + (−0.0553 − 0.349i)8-s + (0.778 + 0.253i)9-s + (−0.705 − 0.0466i)10-s + (−0.297 − 0.914i)11-s + (0.189 − 0.0966i)12-s + (0.605 + 1.18i)13-s + (0.275 − 0.651i)14-s + (−0.104 − 0.412i)15-s + (−0.0772 + 0.237i)16-s + (0.155 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22692 - 0.293887i\)
\(L(\frac12)\) \(\approx\) \(1.22692 - 0.293887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
5 \( 1 + (-2.05 + 0.881i)T \)
7 \( 1 + (-0.185 - 2.63i)T \)
good3 \( 1 + (-0.115 + 0.728i)T + (-2.85 - 0.927i)T^{2} \)
11 \( 1 + (0.985 + 3.03i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.18 - 4.28i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.642 - 4.05i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.04 + 0.759i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.20 + 2.36i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (5.53 + 7.61i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.68 + 3.69i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.85 - 7.57i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-3.77 - 1.22i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (6.47 + 6.47i)T + 43iT^{2} \)
47 \( 1 + (-10.0 - 1.59i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (5.92 + 0.939i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.443 + 1.36i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.66 - 1.84i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.14 - 1.44i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (2.69 - 1.95i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.7 + 6.00i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-5.73 - 7.89i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (15.2 - 2.41i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-1.88 - 5.79i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.01 - 0.795i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.39423054971119607251274048782, −10.40921429678370401791182810932, −9.465610805805435693717054420518, −8.714089874589172970268826546723, −7.963741704304469403078558715893, −6.49627176944413599401239864284, −5.85251468464771566535027255763, −4.30866555540844824117261173003, −2.49852384200429486585027120308, −1.50688504286732478200913166537, 1.40499102891991542770326374476, 3.18303346825824138556128014219, 4.68974425457679234254669989535, 5.79864945739266497915933497816, 7.09427648727150436591632614086, 7.50744739812695152089285667042, 9.025959992628712440290467835385, 9.812066492510673337783597701271, 10.42771547745508707294882658673, 11.00363911928366745252902875901

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