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Properties

Label 2-350-7.4-c1-0-0
Degree $2$
Conductor $350$
Sign $-0.701 + 0.712i$
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 + (−1.5 + 2.59i)3-s + (−0.499 + 0.866i)4-s − 3·6-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−3 − 5.19i)9-s + (−1.50 − 2.59i)12-s + 2·13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (3 − 5.19i)18-s + (1 + 1.73i)19-s + (1.5 − 7.79i)21-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 + 1.49i)3-s + (−0.249 + 0.433i)4-s − 1.22·6-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−1 − 1.73i)9-s + (−0.433 − 0.749i)12-s + 0.554·13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (0.707 − 1.22i)18-s + (0.229 + 0.397i)19-s + (0.327 − 1.70i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.701 + 0.712i$
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: \((2,\ 350,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258116 - 0.615907i\)
\(L(\frac12)\) \(\approx\) \(0.258116 - 0.615907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−12.07437486845391358146796619697, −10.98607620875950455439954139780, −10.26461302921956060053819418277, −9.326217105439490579737873897000, −8.609091153860219118402845834657, −6.95582827991260943650974203612, −5.96941015876984693134168855238, −5.34156699539499772301423828215, −4.13356913470701269888723617041, −3.31740197068921615268708359396, 0.44724677108090400808069923914, 1.97703365690165919918700415792, 3.45868192196357096712005263972, 5.10462796917415574802782240317, 6.16264184187355367482836555314, 6.81380190649926851391580438902, 7.82328455295202986781267255488, 9.164264949147969639751508058194, 10.29299744666972837081543763376, 11.26948205131981146021901079043

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