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Properties

Label 2-350-25.11-c1-0-5
Degree $2$
Conductor $350$
Sign $0.957 + 0.289i$
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  + (−0.809 + 0.587i)2-s + (0.604 − 1.86i)3-s + (0.309 − 0.951i)4-s + (−1.11 + 1.93i)5-s + (0.604 + 1.86i)6-s + 7-s + (0.309 + 0.951i)8-s + (−0.669 − 0.486i)9-s + (−0.233 − 2.22i)10-s + (3.64 − 2.64i)11-s + (−1.58 − 1.14i)12-s + (1.94 + 1.41i)13-s + (−0.809 + 0.587i)14-s + (2.92 + 3.25i)15-s + (−0.809 − 0.587i)16-s + (1.43 + 4.41i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.349 − 1.07i)3-s + (0.154 − 0.475i)4-s + (−0.499 + 0.866i)5-s + (0.246 + 0.759i)6-s + 0.377·7-s + (0.109 + 0.336i)8-s + (−0.223 − 0.162i)9-s + (−0.0739 − 0.703i)10-s + (1.09 − 0.798i)11-s + (−0.456 − 0.331i)12-s + (0.540 + 0.392i)13-s + (−0.216 + 0.157i)14-s + (0.755 + 0.839i)15-s + (−0.202 − 0.146i)16-s + (0.348 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17174 - 0.173027i\)
\(L(\frac12)\) \(\approx\) \(1.17174 - 0.173027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (1.11 - 1.93i)T \)
7 \( 1 - T \)
good3 \( 1 + (-0.604 + 1.86i)T + (-2.42 - 1.76i)T^{2} \)
11 \( 1 + (-3.64 + 2.64i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.94 - 1.41i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.43 - 4.41i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.32 + 7.16i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.09 + 2.97i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.647 + 1.99i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.0580 - 0.178i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.232 - 0.168i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.66 - 4.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + (3.85 - 11.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.99 - 9.22i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.60 + 1.89i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.184 - 0.134i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.96 + 6.04i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.57 + 7.91i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.74 - 2.72i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.224 + 0.690i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.67 - 11.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (11.5 - 8.35i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.37 + 13.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.17540855919439228922337709610, −10.82971885476809208725819249797, −9.273438649962201442165954865425, −8.408575061999906263106312205261, −7.71206856514702477960503004521, −6.63531636534571442024698593187, −6.32076719839583177070131779191, −4.33443820567972878554006516045, −2.78286227863963385629386536153, −1.25226103391635596150390007829, 1.43003907666007575478037591731, 3.49204285877312980561141556321, 4.21905331331114527257351920972, 5.30188909234683977889745940751, 7.04294938126906925150141041895, 8.147807873331568359801590441587, 8.955323431493100456061796532853, 9.605154362364585602748152975386, 10.39751299302132323330346625970, 11.50809612237225182237665058350

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