Properties

Label 2-350-25.9-c1-0-1
Degree $2$
Conductor $350$
Sign $-0.303 - 0.952i$
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 + (−2.21 + 0.719i)3-s + (−0.309 − 0.951i)4-s + (0.562 − 2.16i)5-s + (0.719 − 2.21i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.96 − 1.42i)9-s + (1.42 + 1.72i)10-s + (−1.43 − 1.04i)11-s + (1.36 + 1.88i)12-s + (3.87 + 5.33i)13-s + (−0.809 − 0.587i)14-s + (0.311 + 5.20i)15-s + (−0.809 + 0.587i)16-s + (1.97 + 0.640i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−1.27 + 0.415i)3-s + (−0.154 − 0.475i)4-s + (0.251 − 0.967i)5-s + (0.293 − 0.904i)6-s + 0.377i·7-s + (0.336 + 0.109i)8-s + (0.654 − 0.475i)9-s + (0.449 + 0.546i)10-s + (−0.433 − 0.314i)11-s + (0.395 + 0.544i)12-s + (1.07 + 1.47i)13-s + (−0.216 − 0.157i)14-s + (0.0804 + 1.34i)15-s + (−0.202 + 0.146i)16-s + (0.477 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.303 - 0.952i$
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: \((2,\ 350,\ (\ :1/2),\ -0.303 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354111 + 0.484327i\)
\(L(\frac12)\) \(\approx\) \(0.354111 + 0.484327i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.562 + 2.16i)T \)
7 \( 1 - iT \)
good3 \( 1 + (2.21 - 0.719i)T + (2.42 - 1.76i)T^{2} \)
11 \( 1 + (1.43 + 1.04i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.87 - 5.33i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.97 - 0.640i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.115 + 0.355i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.16 - 7.10i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.90 - 5.84i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.40 - 7.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.895 - 1.23i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.05 + 3.67i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.64iT - 43T^{2} \)
47 \( 1 + (-5.37 + 1.74i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.26 - 1.06i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.4 + 8.34i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-10.3 - 7.55i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.26 + 1.06i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.00 + 6.16i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.12 - 2.92i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.04 - 6.28i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.66 - 2.16i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.79 + 4.93i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (8.40 - 2.73i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.62820741304144296361474622754, −10.87723571787542516921005982888, −9.829325868806297120814638529662, −9.009894291627729001836204919067, −8.167844233448325988018788110341, −6.73120265104576520066772133944, −5.76719309551388863943608524050, −5.25403539499847682003651725112, −4.07714301254550765828462602708, −1.40566782081434311958020849283, 0.61373017537423755166134672036, 2.49959475406481721596772796171, 3.94351219366004154485128844540, 5.59930299790163341864795734882, 6.26967298994087207609987614069, 7.40902710169176418733911648592, 8.236263252186154072499486871240, 9.907975170660249160274486104768, 10.49051955693287586842934183432, 11.05808822418135202622773883647

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