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Properties

Label 2-350-35.33-c1-0-0
Degree $2$
Conductor $350$
Sign $0.175 - 0.984i$
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  + (−0.258 − 0.965i)2-s + (−1.26 − 0.339i)3-s + (−0.866 + 0.499i)4-s + 1.31i·6-s + (−2.49 − 0.884i)7-s + (0.707 + 0.707i)8-s + (−1.10 − 0.637i)9-s + (2.63 + 4.56i)11-s + (1.26 − 0.339i)12-s + (−0.296 + 0.296i)13-s + (−0.209 + 2.63i)14-s + (0.500 − 0.866i)16-s + (−0.896 + 3.34i)17-s + (−0.329 + 1.23i)18-s + (−2.83 + 4.91i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.732 − 0.196i)3-s + (−0.433 + 0.249i)4-s + 0.536i·6-s + (−0.942 − 0.334i)7-s + (0.249 + 0.249i)8-s + (−0.368 − 0.212i)9-s + (0.795 + 1.37i)11-s + (0.366 − 0.0981i)12-s + (−0.0820 + 0.0820i)13-s + (−0.0559 + 0.704i)14-s + (0.125 − 0.216i)16-s + (−0.217 + 0.811i)17-s + (−0.0777 + 0.290i)18-s + (−0.650 + 1.12i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263567 + 0.220836i\)
\(L(\frac12)\) \(\approx\) \(0.263567 + 0.220836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (2.49 + 0.884i)T \)
good3 \( 1 + (1.26 + 0.339i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.296 - 0.296i)T - 13iT^{2} \)
17 \( 1 + (0.896 - 3.34i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.83 - 4.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.89 - 0.776i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.27iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.36 - 5.09i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + (-5.85 - 5.85i)T + 43iT^{2} \)
47 \( 1 + (12.1 - 3.26i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.54 + 9.48i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.19 + 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.4 + 7.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.59 + 0.964i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (9.22 + 2.47i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.66 - 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (-1.97 + 3.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 13.5i)T + 97iT^{2} \)
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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

−11.80300600568343490607206495235, −10.76797683562101980593103291709, −9.949234325336161151684496920846, −9.247678835134348319782218687518, −7.988951364269022911216223216960, −6.71717597004452255006413160603, −6.03679123237643170966769269316, −4.50825221343525068166978600335, −3.48262851152332696360436586767, −1.73971307351220023775841764390, 0.27137996911791011161796677259, 2.97197847290170583467893458928, 4.47165932423807963164990845129, 5.76664605151638981826734983244, 6.21003428682709912800822295297, 7.28440318145443541371801148810, 8.720028071279091919562011437648, 9.150378743857139498372134840445, 10.40603750782611939752901312846, 11.24134619463313199751421147639

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