↑

Properties

Label 2-350-35.3-c1-0-1
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

Related objects

Downloads

Learn more

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: $\chi_{350} (143, \cdot )$
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 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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