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Properties

Label 2-350-35.17-c1-0-7
Degree $2$
Conductor $350$
Sign $0.857 - 0.515i$
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 + (2.94 − 0.788i)3-s + (−0.866 − 0.499i)4-s + 3.04i·6-s + (1.71 + 2.01i)7-s + (0.707 − 0.707i)8-s + (5.43 − 3.13i)9-s + (−1.13 + 1.97i)11-s + (−2.94 − 0.788i)12-s + (−3.37 − 3.37i)13-s + (−2.38 + 1.13i)14-s + (0.500 + 0.866i)16-s + (−0.896 − 3.34i)17-s + (1.62 + 6.06i)18-s + (3.70 + 6.41i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (1.69 − 0.455i)3-s + (−0.433 − 0.249i)4-s + 1.24i·6-s + (0.648 + 0.760i)7-s + (0.249 − 0.249i)8-s + (1.81 − 1.04i)9-s + (−0.342 + 0.594i)11-s + (−0.849 − 0.227i)12-s + (−0.936 − 0.936i)13-s + (−0.638 + 0.303i)14-s + (0.125 + 0.216i)16-s + (−0.217 − 0.811i)17-s + (0.382 + 1.42i)18-s + (0.849 + 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.857 - 0.515i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.857 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95208 + 0.541493i\)
\(L(\frac12)\) \(\approx\) \(1.95208 + 0.541493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 2.01i)T \)
good3 \( 1 + (-2.94 + 0.788i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.13 - 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.37 + 3.37i)T + 13iT^{2} \)
17 \( 1 + (0.896 + 3.34i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.70 - 6.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.89 + 0.776i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.27iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.588 - 2.19i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (-0.512 + 0.512i)T - 43iT^{2} \)
47 \( 1 + (-0.459 - 0.123i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.31 + 12.3i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.08 - 0.627i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.8 - 2.91i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.808 - 0.216i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.66 + 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.888 + 0.888i)T + 83iT^{2} \)
89 \( 1 + (4.56 + 7.91i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 + 1.18i)T - 97iT^{2} \)
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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

−11.87125236465467963822311562923, −10.05998865778343178873276950965, −9.550434574773619104494296027974, −8.492913700257706771221667922202, −7.79814720583309589464009605045, −7.36274547001604938380423764740, −5.79742221161091919546101533109, −4.58607231406184932297528711012, −3.05192753025677375296128214643, −1.92243054662513172381906332488, 1.80933723055707531416643263303, 2.98293181514568086358181187511, 4.03082432542187912798990548809, 4.91466036390575032980347054042, 7.15768833249060584500332868223, 7.86495697093497027199500442057, 8.870974928277401288228969348771, 9.387358209895312776323213536105, 10.43729351470215479034327960633, 11.10772209338705938603520487588

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