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Properties

Label 2-350-7.2-c1-0-5
Degree $2$
Conductor $350$
Sign $0.386 + 0.922i$
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 + (−1 − 1.73i)3-s + (−0.499 − 0.866i)4-s + 1.99·6-s + (2 + 1.73i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)11-s + (−0.999 + 1.73i)12-s + 13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + (−0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s + 0.816·6-s + (0.755 + 0.654i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.452 − 0.783i)11-s + (−0.288 + 0.499i)12-s + 0.277·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.117 − 0.204i)18-s + (0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.719146 - 0.478363i\)
\(L(\frac12)\) \(\approx\) \(0.719146 - 0.478363i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−11.40518483467871710459283141680, −10.55912520901289487820423052030, −9.093523937516463673638436525316, −8.435824106080819055901854354460, −7.40675130525996193179552229267, −6.57699278845753887304651988797, −5.68243447702310916146494104126, −4.70720066656773359075064121070, −2.49972513861462307076670749603, −0.76715960847554012757227895118, 1.70705559868448230388437732704, 3.63828460380210441255045746307, 4.56136019622450354262722794787, 5.41099348820503926504808815869, 7.05308508046181813925036245386, 8.054028167643850306705138259183, 9.146262133486748069002595758059, 10.10720215192894240477848547989, 10.79811574979514652914898104687, 11.17540456799210083939143694180

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