Properties

Label 2-350-7.4-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 + (−0.499 + 0.866i)4-s + (2 − 1.73i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + 5·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (−1.5 + 2.59i)18-s + (2.5 + 4.33i)19-s − 3·22-s + (−3.5 − 6.06i)23-s + (2.5 + 4.33i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.755 − 0.654i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + 1.38·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (−0.353 + 0.612i)18-s + (0.573 + 0.993i)19-s − 0.639·22-s + (−0.729 − 1.26i)23-s + (0.490 + 0.849i)26-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: $\chi_{350} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41764 + 0.942989i\)
\(L(\frac12)\) \(\approx\) \(1.41764 + 0.942989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (8 - 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−11.66323269048498479280085956263, −10.67423866995848936287344453353, −9.990005599916454234639780513615, −8.432844752820524408629777323891, −7.897430831147115771947835116270, −6.95807417259234074157050225892, −5.77708077183459598292002559870, −4.67526715532093570447953523581, −3.84342406883718544027527528705, −1.84383325856557009391926571129, 1.31034123696361728243552015324, 2.96050640500358563979940233413, 4.07877084750982916932396334106, 5.37509321581881459979995080225, 6.18504823601605595008371513547, 7.60537317777427137050936828727, 8.792905082338304325384859022469, 9.388118800271014754235695322422, 10.68177613968475580988606415535, 11.43037689665071350163098355732

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