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Properties

Label 2-350-35.3-c1-0-8
Degree $2$
Conductor $350$
Sign $0.862 - 0.506i$
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 + (0.189 + 0.707i)3-s + (0.866 + 0.499i)4-s + 0.732i·6-s + (2.19 − 1.48i)7-s + (0.707 + 0.707i)8-s + (2.13 − 1.23i)9-s + (0.633 − 1.09i)11-s + (−0.189 + 0.707i)12-s + (−4.38 + 4.38i)13-s + (2.49 − 0.866i)14-s + (0.500 + 0.866i)16-s + (0.448 − 0.120i)17-s + (2.38 − 0.637i)18-s + (1.46 + 1.26i)21-s + (0.896 − 0.896i)22-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.109 + 0.408i)3-s + (0.433 + 0.249i)4-s + 0.298i·6-s + (0.827 − 0.560i)7-s + (0.249 + 0.249i)8-s + (0.711 − 0.410i)9-s + (0.191 − 0.331i)11-s + (−0.0546 + 0.204i)12-s + (−1.21 + 1.21i)13-s + (0.668 − 0.231i)14-s + (0.125 + 0.216i)16-s + (0.108 − 0.0291i)17-s + (0.561 − 0.150i)18-s + (0.319 + 0.276i)21-s + (0.191 − 0.191i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.862 - 0.506i$
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: \((2,\ 350,\ (\ :1/2),\ 0.862 - 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11777 + 0.575780i\)
\(L(\frac12)\) \(\approx\) \(2.11777 + 0.575780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.189 - 0.707i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.38 - 4.38i)T - 13iT^{2} \)
17 \( 1 + (-0.448 + 0.120i)T + (14.7 - 8.5i)T^{2} \)
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 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + (7.58 + 7.58i)T + 43iT^{2} \)
47 \( 1 + (1.01 - 3.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.91 + 2.12i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.09 - 12.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.29 - 3.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.984 + 3.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (2.07 + 7.72i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.86 - 5.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 + (-5.59 - 9.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.05 + 8.05i)T + 97iT^{2} \)
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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

−11.77713329456370836223716829761, −10.71176907576795315346950167670, −9.798806795685553265745853581536, −8.816086911526207407272166732964, −7.45956368869752446318080767888, −6.92608083449093919524776581396, −5.45624584660090289418466164813, −4.45518323177298548139770402867, −3.70531737237727504926142500809, −1.88160319861701577871937294270, 1.72216218698676265093020740293, 2.90354009244970516832302213355, 4.61276775905407458534392275834, 5.22286353759257010671457182826, 6.58666597753176709300893634717, 7.59490810583609116629981723394, 8.350827886013622903589726919765, 9.827540318987451211346923758087, 10.55152888336040606578288712215, 11.64549297132054079410815090019

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