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Properties

Label 2-350-35.12-c1-0-9
Degree $2$
Conductor $350$
Sign $-0.914 + 0.403i$
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 + (0.752 − 2.80i)3-s + (0.866 − 0.499i)4-s + 2.90i·6-s + (−2.58 − 0.559i)7-s + (−0.707 + 0.707i)8-s + (−4.71 − 2.72i)9-s + (−1.83 − 3.17i)11-s + (−0.752 − 2.80i)12-s + (0.830 + 0.830i)13-s + (2.64 − 0.128i)14-s + (0.500 − 0.866i)16-s + (0.761 + 0.204i)17-s + (5.26 + 1.41i)18-s + (−1.09 + 1.89i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.434 − 1.62i)3-s + (0.433 − 0.249i)4-s + 1.18i·6-s + (−0.977 − 0.211i)7-s + (−0.249 + 0.249i)8-s + (−1.57 − 0.908i)9-s + (−0.553 − 0.958i)11-s + (−0.217 − 0.810i)12-s + (0.230 + 0.230i)13-s + (0.706 − 0.0343i)14-s + (0.125 − 0.216i)16-s + (0.184 + 0.0494i)17-s + (1.24 + 0.332i)18-s + (−0.251 + 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.914 + 0.403i$
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: \((2,\ 350,\ (\ :1/2),\ -0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155578 - 0.738163i\)
\(L(\frac12)\) \(\approx\) \(0.155578 - 0.738163i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.559i)T \)
good3 \( 1 + (-0.752 + 2.80i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.83 + 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.830 - 0.830i)T + 13iT^{2} \)
17 \( 1 + (-0.761 - 0.204i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.09 - 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.21 + 4.54i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.62iT - 29T^{2} \)
31 \( 1 + (-0.0359 + 0.0207i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.248 + 0.0664i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (-0.474 + 0.474i)T - 43iT^{2} \)
47 \( 1 + (1.65 + 6.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-7.64 - 2.04i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.35 + 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 0.996i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 6.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (2.55 - 9.52i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-11.6 - 6.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.73 + 9.73i)T + 83iT^{2} \)
89 \( 1 + (0.715 - 1.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 - 3.16i)T - 97iT^{2} \)
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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

−11.06215664302613888415425848979, −10.10051521475689362526051624452, −8.843640745090900963681370820202, −8.277055506476088629553558995159, −7.29394064986006469574918526073, −6.54600045604718691377006803819, −5.76100106631222054406981749382, −3.39980276992142838621981821395, −2.20552756568883915565992309303, −0.59060824774160595215302928275, 2.59408919340840467352756228064, 3.59370450782677727160822614031, 4.75647410649678481203173822274, 6.00224162368138015453952607493, 7.41566874479596305602280107848, 8.505789308288415413727058877871, 9.458675577590509153666344232821, 9.853525248947261751634102220344, 10.57656414274936077976292326711, 11.54301675395138167623147780265

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