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Properties

Label 2-350-35.9-c3-0-0
Degree $2$
Conductor $350$
Sign $-0.111 + 0.993i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (−5.99 + 3.46i)3-s + (1.99 + 3.46i)4-s + 13.8·6-s + (−12.7 + 13.4i)7-s − 7.99i·8-s + (10.4 − 18.1i)9-s + (−18.6 − 32.3i)11-s + (−23.9 − 13.8i)12-s + 89.1i·13-s + (35.5 − 10.5i)14-s + (−8 + 13.8i)16-s + (−60.7 + 35.0i)17-s + (−36.2 + 20.9i)18-s + (−62.7 + 108. i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.15 + 0.666i)3-s + (0.249 + 0.433i)4-s + 0.942·6-s + (−0.688 + 0.725i)7-s − 0.353i·8-s + (0.387 − 0.671i)9-s + (−0.512 − 0.886i)11-s + (−0.577 − 0.333i)12-s + 1.90i·13-s + (0.677 − 0.200i)14-s + (−0.125 + 0.216i)16-s + (−0.866 + 0.500i)17-s + (−0.475 + 0.274i)18-s + (−0.757 + 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.111 + 0.993i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -0.111 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.002960341874\)
\(L(\frac12)\) \(\approx\) \(0.002960341874\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
5 \( 1 \)
7 \( 1 + (12.7 - 13.4i)T \)
good3 \( 1 + (5.99 - 3.46i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (18.6 + 32.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 89.1iT - 2.19e3T^{2} \)
17 \( 1 + (60.7 - 35.0i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (62.7 - 108. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (5.42 + 3.13i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 95.4T + 2.43e4T^{2} \)
31 \( 1 + (-108. - 188. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (182. + 105. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 12.2T + 6.89e4T^{2} \)
43 \( 1 - 131. iT - 7.95e4T^{2} \)
47 \( 1 + (315. + 182. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-431. + 249. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (179. + 310. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-77.3 + 133. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (58.4 - 33.7i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 517.T + 3.57e5T^{2} \)
73 \( 1 + (-584. + 337. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-465. + 806. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.43e3iT - 5.71e5T^{2} \)
89 \( 1 + (168. - 291. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 179. iT - 9.12e5T^{2} \)
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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

−11.65876784545969964724365767216, −10.71254127058286212331323997045, −10.12916293755177756745380997928, −9.050354155689729950353853089859, −8.369374773309781790448949163077, −6.61818885598831995949376305612, −6.11623345481482427516861809141, −4.82362222613093034826871144399, −3.63883025549874068251927509625, −2.01870807918413291602486797764, 0.00215216819112472543976515851, 0.78042451588569298422952682757, 2.67076128106222606406040634692, 4.66418772315135885069756686377, 5.70433273715271204607833129347, 6.71869238471266691893017288679, 7.23715357502155519397964834283, 8.297258988744657609201172010400, 9.645634319129849596051596357332, 10.48928059625314765543463197594

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