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Properties

Label 2-350-7.4-c1-0-4
Degree $2$
Conductor $350$
Sign $-0.266 - 0.963i$
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 + (−0.5 + 2.59i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1 − 1.73i)11-s + (−2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−3.5 + 6.06i)17-s + (−1.5 + 2.59i)18-s + 1.99·22-s + (1.5 + 2.59i)23-s + (−2 − 1.73i)28-s + 6·29-s + (3.5 − 6.06i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.301 − 0.522i)11-s + (−0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.848 + 1.47i)17-s + (−0.353 + 0.612i)18-s + 0.426·22-s + (0.312 + 0.541i)23-s + (−0.377 − 0.327i)28-s + 1.11·29-s + (0.628 − 1.08i)31-s + (0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 350,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901524 + 1.18503i\)
\(L(\frac12)\) \(\approx\) \(0.901524 + 1.18503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−11.86829101099404790262712064987, −10.91873840622636396596505910858, −9.828613902065009203468454290904, −8.675316991609847082404949933469, −8.116192448671307693468838603486, −6.80991542849221230349033387897, −5.96132472153781885865701835896, −4.96516076979752092931154537741, −3.76230548808931036220762617250, −2.20496407227810848497295250088, 1.00315219492347088514687749877, 2.85248647170399003682824530588, 4.10702267231352623105337212255, 4.88489845734171203037522675899, 6.60121912851246192474476526756, 7.07327476270978309141219495726, 8.646716636315804039807457408765, 9.656785970563092861908815034313, 10.26103845395232419857560481787, 11.30186564908633296258363199411

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