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Properties

Label 2-350-35.33-c1-0-11
Degree $2$
Conductor $350$
Sign $-0.977 + 0.211i$
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 + (0.279 + 0.0749i)3-s + (−0.866 + 0.499i)4-s − 0.289i·6-s + (−2.64 + 0.126i)7-s + (0.707 + 0.707i)8-s + (−2.52 − 1.45i)9-s + (−2.81 − 4.87i)11-s + (−0.279 + 0.0749i)12-s + (−1.42 + 1.42i)13-s + (0.806 + 2.51i)14-s + (0.500 − 0.866i)16-s + (1.37 − 5.12i)17-s + (−0.754 + 2.81i)18-s + (1.94 − 3.37i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.161 + 0.0432i)3-s + (−0.433 + 0.249i)4-s − 0.118i·6-s + (−0.998 + 0.0477i)7-s + (0.249 + 0.249i)8-s + (−0.841 − 0.486i)9-s + (−0.848 − 1.46i)11-s + (−0.0807 + 0.0216i)12-s + (−0.396 + 0.396i)13-s + (0.215 + 0.673i)14-s + (0.125 − 0.216i)16-s + (0.333 − 1.24i)17-s + (−0.177 + 0.663i)18-s + (0.446 − 0.773i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.977 + 0.211i$
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: \((2,\ 350,\ (\ :1/2),\ -0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0595395 - 0.557488i\)
\(L(\frac12)\) \(\approx\) \(0.0595395 - 0.557488i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.126i)T \)
good3 \( 1 + (-0.279 - 0.0749i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.81 + 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.42 - 1.42i)T - 13iT^{2} \)
17 \( 1 + (-1.37 + 5.12i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.94 + 3.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 - 0.290i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 + (3.33 - 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.30 - 4.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.21iT - 41T^{2} \)
43 \( 1 + (1.85 + 1.85i)T + 43iT^{2} \)
47 \( 1 + (-5.69 + 1.52i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.357 + 1.33i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.73 + 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 + 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.816 - 0.218i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (5.42 + 1.45i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.41 + 3.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.67 + 5.67i)T - 83iT^{2} \)
89 \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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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

−11.19566032535502428053727733522, −10.08674050814731912911192075100, −9.245732956990494348013783942055, −8.587322304129534488439517814614, −7.35690550473229429234839169331, −6.10586733934832848145024716046, −5.05767420335229920442073142452, −3.30771278537313678374261376173, −2.83370557408772255471332660163, −0.38018348880425589581256190081, 2.36031563221958553325250528746, 3.87937846301833558649253379256, 5.30798584101102816543717663812, 6.08211565074860465971103747515, 7.39653681987755263958933692775, 7.924264645093433506915744764000, 9.114293832507225461771633507734, 10.03633301246677338237774377935, 10.61051000952306453636471042862, 12.21730386061287380853366328822

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