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Properties

Label 2-350-25.11-c1-0-6
Degree $2$
Conductor $350$
Sign $0.974 + 0.223i$
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  + (0.809 − 0.587i)2-s + (−0.0727 + 0.223i)3-s + (0.309 − 0.951i)4-s + (1.97 + 1.05i)5-s + (0.0727 + 0.223i)6-s + 7-s + (−0.309 − 0.951i)8-s + (2.38 + 1.73i)9-s + (2.21 − 0.308i)10-s + (−2.81 + 2.04i)11-s + (0.190 + 0.138i)12-s + (−0.624 − 0.453i)13-s + (0.809 − 0.587i)14-s + (−0.378 + 0.365i)15-s + (−0.809 − 0.587i)16-s + (0.149 + 0.461i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.0419 + 0.129i)3-s + (0.154 − 0.475i)4-s + (0.882 + 0.470i)5-s + (0.0296 + 0.0913i)6-s + 0.377·7-s + (−0.109 − 0.336i)8-s + (0.794 + 0.576i)9-s + (0.700 − 0.0976i)10-s + (−0.850 + 0.617i)11-s + (0.0549 + 0.0399i)12-s + (−0.173 − 0.125i)13-s + (0.216 − 0.157i)14-s + (−0.0978 + 0.0942i)15-s + (−0.202 − 0.146i)16-s + (0.0363 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.974 + 0.223i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.974 + 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07114 - 0.234668i\)
\(L(\frac12)\) \(\approx\) \(2.07114 - 0.234668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-1.97 - 1.05i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.0727 - 0.223i)T + (-2.42 - 1.76i)T^{2} \)
11 \( 1 + (2.81 - 2.04i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.624 + 0.453i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.149 - 0.461i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.01 + 6.19i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.90 + 5.01i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.84 - 5.67i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.20 + 3.70i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.06 + 0.771i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.86 + 4.26i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.85T + 43T^{2} \)
47 \( 1 + (3.48 - 10.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.66 + 8.19i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.49 + 6.17i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.5 - 7.68i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.45 - 7.55i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.0996 - 0.306i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.07 + 4.41i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.331 + 1.01i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.48 + 4.58i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (9.89 - 7.19i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.340 - 1.04i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.17664239380288597393049164230, −10.65571998262931892190266306921, −9.932191579240851452167580248816, −8.883621505369860069916044167598, −7.39197775714332095193671951182, −6.63766219914518756767028562470, −5.18336615289120984671745298684, −4.69342025450636131201183147132, −2.92039649976911951959147763655, −1.87237632033751584356684004171, 1.67215163130949558656185277572, 3.36135940048854413754041389601, 4.76595201407456083366681378153, 5.60458057141602711228075215176, 6.54549013747095900793577831668, 7.65530032459656092763198100937, 8.615992524114451298664582237371, 9.679380429316541689948644211811, 10.55666150734487870773291720373, 11.76061377254652771567690202350

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