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Properties

Label 2-350-35.4-c3-0-15
Degree $2$
Conductor $350$
Sign $0.726 + 0.687i$
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 + (−3.82 − 2.20i)3-s + (1.99 − 3.46i)4-s + 8.83·6-s + (18.4 + 2.04i)7-s + 7.99i·8-s + (−3.75 − 6.49i)9-s + (18.5 − 32.1i)11-s + (−15.2 + 8.83i)12-s + 47.2i·13-s + (−33.9 + 14.8i)14-s + (−8 − 13.8i)16-s + (32.7 + 18.8i)17-s + (12.9 + 7.50i)18-s + (3.80 + 6.58i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.735 − 0.424i)3-s + (0.249 − 0.433i)4-s + 0.600·6-s + (0.993 + 0.110i)7-s + 0.353i·8-s + (−0.138 − 0.240i)9-s + (0.508 − 0.880i)11-s + (−0.367 + 0.212i)12-s + 1.00i·13-s + (−0.647 + 0.283i)14-s + (−0.125 − 0.216i)16-s + (0.466 + 0.269i)17-s + (0.170 + 0.0982i)18-s + (0.0459 + 0.0795i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.092329198\)
\(L(\frac12)\) \(\approx\) \(1.092329198\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 - 2.04i)T \)
good3 \( 1 + (3.82 + 2.20i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-18.5 + 32.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 47.2iT - 2.19e3T^{2} \)
17 \( 1 + (-32.7 - 18.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.80 - 6.58i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (128. - 74.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 84.5T + 2.43e4T^{2} \)
31 \( 1 + (-64.9 + 112. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-128. + 74.0i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 6.16T + 6.89e4T^{2} \)
43 \( 1 + 523. iT - 7.95e4T^{2} \)
47 \( 1 + (-367. + 211. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (298. + 172. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-330. + 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (5.95 + 10.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-624. - 360. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 161.T + 3.57e5T^{2} \)
73 \( 1 + (-526. - 303. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-107. - 186. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 390. iT - 5.71e5T^{2} \)
89 \( 1 + (756. + 1.31e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 853. iT - 9.12e5T^{2} \)
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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

−11.22800076857922112469329483397, −9.995498582275742488841890324014, −8.922498136479315259255421674459, −8.167384864236154584834713309478, −7.09213959869364912679392687141, −6.14042088691948296297283819776, −5.41370721602377852476694963408, −3.91278629059299993686635433097, −1.88258551285822707650054001365, −0.66953796567052817744492676452, 1.03190208527710728906445769705, 2.52812364501675135111459564665, 4.25532051450051622600745546358, 5.12519392405725501029219550663, 6.31737992780110351255081048079, 7.69051328561503128369735936925, 8.262743731997371586998085453150, 9.593181409562649249572735725976, 10.37252723405317141743264381982, 11.00998850873281962444143654862

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