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Properties

Label 2-350-35.3-c1-0-2
Degree $2$
Conductor $350$
Sign $-0.323 - 0.946i$
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  + (0.965 + 0.258i)2-s + (0.707 + 2.63i)3-s + (0.866 + 0.499i)4-s + 2.73i·6-s + (−2.19 + 1.48i)7-s + (0.707 + 0.707i)8-s + (−3.86 + 2.23i)9-s + (2.36 − 4.09i)11-s + (−0.707 + 2.63i)12-s + (−2.96 + 2.96i)13-s + (−2.49 + 0.866i)14-s + (0.500 + 0.866i)16-s + (6.24 − 1.67i)17-s + (−4.31 + 1.15i)18-s + (−5.46 − 4.73i)21-s + (3.34 − 3.34i)22-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.408 + 1.52i)3-s + (0.433 + 0.249i)4-s + 1.11i·6-s + (−0.827 + 0.560i)7-s + (0.249 + 0.249i)8-s + (−1.28 + 0.744i)9-s + (0.713 − 1.23i)11-s + (−0.204 + 0.761i)12-s + (−0.822 + 0.822i)13-s + (−0.668 + 0.231i)14-s + (0.125 + 0.216i)16-s + (1.51 − 0.405i)17-s + (−1.01 + 0.272i)18-s + (−1.19 − 1.03i)21-s + (0.713 − 0.713i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20566 + 1.68662i\)
\(L(\frac12)\) \(\approx\) \(1.20566 + 1.68662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.19 - 1.48i)T \)
good3 \( 1 + (-0.707 - 2.63i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.36 + 4.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.96 - 2.96i)T - 13iT^{2} \)
17 \( 1 + (-6.24 + 1.67i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.34 + 5.01i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.69 - 1.79i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.803iT - 41T^{2} \)
43 \( 1 + (5.13 + 5.13i)T + 43iT^{2} \)
47 \( 1 + (2.56 - 9.58i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.12 - 0.568i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.90 + 3.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.29 + 5.36i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.67 + 13.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 + (-2.07 - 7.72i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.13 + 4.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.24 + 4.24i)T - 83iT^{2} \)
89 \( 1 + (0.401 + 0.696i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.64 + 6.64i)T + 97iT^{2} \)
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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

−11.76244946427109909815220506570, −10.84095237009677411992339155771, −9.756996996027679398418434056540, −9.228657856699366857864039161124, −8.231382726349252900680520253057, −6.71461052909782243212101576269, −5.68666155607968127325653564042, −4.71426913123863726967518120315, −3.58960659016036590004831415659, −2.88384053864672112144041763169, 1.28195923910432348071736435952, 2.66519821427633086236573126017, 3.82154750463146443752166614797, 5.44425194074128028417026060625, 6.56497434763520419617288826523, 7.30887954790642343076715343841, 7.906461569392576606758637229264, 9.550214094352869434224485292061, 10.20625326697458291977038824239, 11.77006506921211224298667556350

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