Properties

Label 2-350-7.4-c1-0-6
Degree $2$
Conductor $350$
Sign $0.605 + 0.795i$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (0.5 + 2.59i)7-s + 0.999·8-s + (1 + 1.73i)9-s + (3 − 5.19i)11-s + (0.499 + 0.866i)12-s + 4·13-s + (2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)18-s + (−1 − 1.73i)19-s + (2.5 + 0.866i)21-s − 6·22-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (0.188 + 0.981i)7-s + 0.353·8-s + (0.333 + 0.577i)9-s + (0.904 − 1.56i)11-s + (0.144 + 0.249i)12-s + 1.10·13-s + (0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.235 − 0.408i)18-s + (−0.229 − 0.397i)19-s + (0.545 + 0.188i)21-s − 1.27·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19460 - 0.592199i\)
\(L(\frac12)\) \(\approx\) \(1.19460 - 0.592199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (8 - 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−11.14633118662792361884377904949, −10.80464514412187505352138054054, −9.189683285765849672773553132719, −8.708747181862142582449605468436, −7.918198123693549389279408758495, −6.56068623140851519642977026679, −5.53934080598421419226546320249, −3.95193641127438853822237163575, −2.71238625560086399033449734165, −1.37271047232170476058201585725, 1.46375846141625953745599028381, 3.87121871979297141974644355668, 4.36038292048665956218834927761, 5.99059187620313463522781212705, 6.99614545356602163056626935680, 7.73052133363762198184662013333, 9.003532390468188680012755658569, 9.646881200135278145608348813606, 10.40390860085647571050119551459, 11.44531500776178172422114943476

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