Properties

Label 2-350-35.12-c1-0-1
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 + (−0.189 + 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

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

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: $\chi_{350} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739534 + 0.528645i\)
\(L(\frac12)\) \(\approx\) \(0.739534 + 0.528645i\)
\(L(\frac{3}{2})\) 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

−11.23350823321948040586110581839, −10.75146503007382649428151425214, −9.548530330765114682701347799985, −9.291171206997215874244594238308, −7.85853140665851520432019437493, −6.94340983288125505622793124801, −6.14084843364540490287464073544, −4.60280075019561125029481541919, −3.52375830646315704656562757256, −1.55709130607474577205702890026, 0.891258762398830228749968718781, 2.69015214770473995984757735314, 3.91565847012070876304489535727, 5.85383431564420040507780175083, 6.46877394669575187217376199387, 7.62082939314298613936116739015, 8.601397536418466029779356806347, 9.414651987438203570669663798777, 10.36223643465362091599265339853, 11.17073736941514573459221037302

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