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Properties

Label 2-350-175.83-c1-0-8
Degree $2$
Conductor $350$
Sign $-0.147 - 0.989i$
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  + (0.891 + 0.453i)2-s + (−0.317 + 2.00i)3-s + (0.587 + 0.809i)4-s + (1.97 + 1.03i)5-s + (−1.19 + 1.64i)6-s + (2.06 + 1.65i)7-s + (0.156 + 0.987i)8-s + (−1.06 − 0.344i)9-s + (1.29 + 1.82i)10-s + (−1.97 − 6.07i)11-s + (−1.80 + 0.921i)12-s + (−1.86 − 3.65i)13-s + (1.08 + 2.41i)14-s + (−2.71 + 3.63i)15-s + (−0.309 + 0.951i)16-s + (0.583 + 3.68i)17-s + ⋯
L(s)  = 1  + (0.630 + 0.321i)2-s + (−0.183 + 1.15i)3-s + (0.293 + 0.404i)4-s + (0.885 + 0.464i)5-s + (−0.486 + 0.670i)6-s + (0.780 + 0.625i)7-s + (0.0553 + 0.349i)8-s + (−0.353 − 0.114i)9-s + (0.408 + 0.577i)10-s + (−0.595 − 1.83i)11-s + (−0.521 + 0.265i)12-s + (−0.516 − 1.01i)13-s + (0.290 + 0.644i)14-s + (−0.699 + 0.939i)15-s + (−0.0772 + 0.237i)16-s + (0.141 + 0.893i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36457 + 1.58318i\)
\(L(\frac12)\) \(\approx\) \(1.36457 + 1.58318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 - 0.453i)T \)
5 \( 1 + (-1.97 - 1.03i)T \)
7 \( 1 + (-2.06 - 1.65i)T \)
good3 \( 1 + (0.317 - 2.00i)T + (-2.85 - 0.927i)T^{2} \)
11 \( 1 + (1.97 + 6.07i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.86 + 3.65i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.583 - 3.68i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (4.00 + 2.90i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.968 + 1.90i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-2.28 - 3.14i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.56 + 3.53i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.834 + 1.63i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (9.86 + 3.20i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (7.82 + 7.82i)T + 43iT^{2} \)
47 \( 1 + (2.56 + 0.406i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-11.9 - 1.89i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.397 + 1.22i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.46 + 0.800i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-12.5 + 1.98i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (5.66 - 4.11i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.88 - 3.50i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-6.06 - 8.35i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.78 - 0.916i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (0.170 + 0.523i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-9.85 - 1.56i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.52128638061681268575743428966, −10.63951900407391688946335837479, −10.31762349011738670783896240458, −8.827967768515326720268714111882, −8.195693242138116529397042347221, −6.59896398456591519765724564366, −5.45349529554260099560312121963, −5.19194146875005370233213510912, −3.64276652122189294626817028916, −2.51256270875316050235281269187, 1.54224429538795291942873086972, 2.21494095148152342101114655236, 4.49060323635084188369881691980, 5.07757445172634541764076142666, 6.58093463757694393403177648312, 7.12188232647489875969668506718, 8.187459002448023792991334953939, 9.742752079461050528892331689302, 10.23013968534492509641108174871, 11.70761989271223004222482998559

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