Properties

Label 2-350-25.9-c1-0-15
Degree $2$
Conductor $350$
Sign $-0.904 + 0.427i$
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  + (0.587 − 0.809i)2-s + (−0.331 + 0.107i)3-s + (−0.309 − 0.951i)4-s + (−1.25 − 1.84i)5-s + (−0.107 + 0.331i)6-s i·7-s + (−0.951 − 0.309i)8-s + (−2.32 + 1.69i)9-s + (−2.23 − 0.0683i)10-s + (−2.18 − 1.58i)11-s + (0.204 + 0.281i)12-s + (−2.81 − 3.87i)13-s + (−0.809 − 0.587i)14-s + (0.616 + 0.477i)15-s + (−0.809 + 0.587i)16-s + (5.80 + 1.88i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.191 + 0.0621i)3-s + (−0.154 − 0.475i)4-s + (−0.562 − 0.826i)5-s + (−0.0439 + 0.135i)6-s − 0.377i·7-s + (−0.336 − 0.109i)8-s + (−0.776 + 0.563i)9-s + (−0.706 − 0.0216i)10-s + (−0.658 − 0.478i)11-s + (0.0591 + 0.0813i)12-s + (−0.781 − 1.07i)13-s + (−0.216 − 0.157i)14-s + (0.159 + 0.123i)15-s + (−0.202 + 0.146i)16-s + (1.40 + 0.457i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.904 + 0.427i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210387 - 0.937491i\)
\(L(\frac12)\) \(\approx\) \(0.210387 - 0.937491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (1.25 + 1.84i)T \)
7 \( 1 + iT \)
good3 \( 1 + (0.331 - 0.107i)T + (2.42 - 1.76i)T^{2} \)
11 \( 1 + (2.18 + 1.58i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.81 + 3.87i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.80 - 1.88i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.15 + 3.55i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.879 - 1.21i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.88 + 8.87i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.33 - 4.12i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.24 + 3.08i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.11 + 6.62i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.26iT - 43T^{2} \)
47 \( 1 + (-10.5 + 3.42i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.496 + 0.161i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.82 + 2.05i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.73 + 1.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-13.2 - 4.32i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.84 + 8.75i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.88 - 5.34i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.21 - 9.88i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-7.89 - 2.56i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.96 + 1.42i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.32 + 1.08i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.13532167976840428567117379547, −10.42129998483292929599016467266, −9.395002371497594382558263590497, −8.151169066277423777746915136123, −7.58881789940893121098226579271, −5.62773593136791638673911492593, −5.23465970531273914779749794043, −3.90291411890134503860257316053, −2.67915729243788550445732864721, −0.58221858314141814870849736676, 2.65708018560552357632620116662, 3.76929682212897332052742671633, 5.15084845245999202563450735970, 6.07925008685909663534592661313, 7.17367790181059761872999630954, 7.77756387576551526672574140200, 9.022532477408984044559373417098, 10.01947036892220071686843989163, 11.18805512013304112256719389609, 12.10321042144166667808053982081

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