Properties

Label 2-350-7.4-c3-0-23
Degree $2$
Conductor $350$
Sign $0.386 - 0.922i$
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 + 1.73i)2-s + (−2.5 + 4.33i)3-s + (−1.99 + 3.46i)4-s − 10·6-s + (14 − 12.1i)7-s − 7.99·8-s + (0.999 + 1.73i)9-s + (28.5 − 49.3i)11-s + (−10 − 17.3i)12-s + 70·13-s + (35 + 12.1i)14-s + (−8 − 13.8i)16-s + (25.5 − 44.1i)17-s + (−1.99 + 3.46i)18-s + (−2.5 − 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.481 + 0.833i)3-s + (−0.249 + 0.433i)4-s − 0.680·6-s + (0.755 − 0.654i)7-s − 0.353·8-s + (0.0370 + 0.0641i)9-s + (0.781 − 1.35i)11-s + (−0.240 − 0.416i)12-s + 1.49·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.0261 + 0.0453i)18-s + (−0.0301 − 0.0522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.299018778\)
\(L(\frac12)\) \(\approx\) \(2.299018778\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-14 + 12.1i)T \)
good3 \( 1 + (2.5 - 4.33i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 + (-25.5 + 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 - 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 114T + 2.43e4T^{2} \)
31 \( 1 + (11.5 - 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (126.5 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 - 124T + 7.95e4T^{2} \)
47 \( 1 + (-100.5 - 174. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (196.5 - 340. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (109.5 - 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-354.5 - 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-209.5 + 362. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 96T + 3.57e5T^{2} \)
73 \( 1 + (156.5 - 271. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (230.5 + 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.83e3T + 9.12e5T^{2} \)
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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

−11.10120509860722805764954056191, −10.60501630831194219221665983919, −9.238108530593104452575772928186, −8.418548346751709060095335691810, −7.38534328075704442680927363536, −6.16125339040493935205267338272, −5.35233417248031440227197322745, −4.26564316463339545885994796397, −3.47558361068407025944971424801, −1.03866875286324919153150668894, 1.16699332580538066909847275053, 1.95825256698028243988393907467, 3.72126794100720122011573819414, 4.86992004788935374197038202736, 6.07426615345231567469040801933, 6.78553316966676088688357338147, 8.117134297795076286777946332493, 9.050606310313651037760441213752, 10.16374787402305466313065997166, 11.21307607014289187109037072916

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