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Properties

Label 2-350-35.18-c2-0-22
Degree $2$
Conductor $350$
Sign $0.955 + 0.295i$
Analytic cond. $9.53680$
Root an. cond. $3.08817$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (4.37 − 1.17i)3-s + (1.73 + i)4-s + 6.40·6-s + (−4.40 − 5.44i)7-s + (1.99 + 2i)8-s + (9.94 − 5.74i)9-s + (3.27 − 5.68i)11-s + (8.74 + 2.34i)12-s + (9.54 + 9.54i)13-s + (−4.01 − 9.04i)14-s + (1.99 + 3.46i)16-s + (−0.550 − 2.05i)17-s + (15.6 − 4.20i)18-s + (23.5 − 13.5i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (1.45 − 0.390i)3-s + (0.433 + 0.250i)4-s + 1.06·6-s + (−0.628 − 0.777i)7-s + (0.249 + 0.250i)8-s + (1.10 − 0.637i)9-s + (0.298 − 0.516i)11-s + (0.728 + 0.195i)12-s + (0.734 + 0.734i)13-s + (−0.286 − 0.646i)14-s + (0.124 + 0.216i)16-s + (−0.0323 − 0.120i)17-s + (0.871 − 0.233i)18-s + (1.23 − 0.714i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ 0.955 + 0.295i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.81647 - 0.576198i\)
\(L(\frac12)\) \(\approx\) \(3.81647 - 0.576198i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
5 \( 1 \)
7 \( 1 + (4.40 + 5.44i)T \)
good3 \( 1 + (-4.37 + 1.17i)T + (7.79 - 4.5i)T^{2} \)
11 \( 1 + (-3.27 + 5.68i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-9.54 - 9.54i)T + 169iT^{2} \)
17 \( 1 + (0.550 + 2.05i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-23.5 + 13.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-0.797 + 2.97i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 52.6iT - 841T^{2} \)
31 \( 1 + (19.3 - 33.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (55.0 + 14.7i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 75.4T + 1.68e3T^{2} \)
43 \( 1 + (38.7 + 38.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (16.9 + 4.53i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-54.5 + 14.6i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (67.7 + 39.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-48.2 - 83.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.10 - 22.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 117.T + 5.04e3T^{2} \)
73 \( 1 + (-23.6 + 6.34i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (22.3 - 12.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (31.8 + 31.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (27.4 - 15.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-52.3 + 52.3i)T - 9.40e3iT^{2} \)
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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

−11.35130813896129453540844371467, −10.25980125874257059868315935149, −9.038573642716220330570044790329, −8.516919749574733511287048327594, −7.05986398777203027804231942616, −6.90336353542710400955513550665, −5.19251334290484634854649652744, −3.60740894071145182345810544283, −3.25678128348834207510619862316, −1.56826558962089577014493463772, 1.97439624685500789754740566738, 3.17944469145099210379524586026, 3.79361858950352546202740580470, 5.27072567359029689582640198881, 6.37178241014992745800875083186, 7.72361788973540412960950946994, 8.532204219857420817267529563042, 9.629293135533987631017566760112, 10.03633451405913929864091522594, 11.51200444167615258926302645496

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