↑

Properties

Label 2-370-5.4-c1-0-1
Degree $2$
Conductor $370$
Sign $-0.635 + 0.771i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + i·2-s + 2.72i·3-s − 4-s + (−1.42 + 1.72i)5-s − 2.72·6-s − 4.14i·7-s − i·8-s − 4.45·9-s + (−1.72 − 1.42i)10-s − 4.76·11-s − 2.72i·12-s + 3.91i·13-s + 4.14·14-s + (−4.71 − 3.88i)15-s + 16-s + 3.31i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.57i·3-s − 0.5·4-s + (−0.635 + 0.771i)5-s − 1.11·6-s − 1.56i·7-s − 0.353i·8-s − 1.48·9-s + (−0.545 − 0.449i)10-s − 1.43·11-s − 0.788i·12-s + 1.08i·13-s + 1.10·14-s + (−1.21 − 1.00i)15-s + 0.250·16-s + 0.804i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.635 + 0.771i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.635 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278007 - 0.589096i\)
\(L(\frac12)\) \(\approx\) \(0.278007 - 0.589096i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.42 - 1.72i)T \)
37 \( 1 - iT \)
good3 \( 1 - 2.72iT - 3T^{2} \)
7 \( 1 + 4.14iT - 7T^{2} \)
11 \( 1 + 4.76T + 11T^{2} \)
13 \( 1 - 3.91iT - 13T^{2} \)
17 \( 1 - 3.31iT - 17T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 - 1.54iT - 23T^{2} \)
29 \( 1 + 8.87T + 29T^{2} \)
31 \( 1 - 9.75T + 31T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
43 \( 1 - 9.99iT - 43T^{2} \)
47 \( 1 - 4.82iT - 47T^{2} \)
53 \( 1 - 5.13iT - 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 + 4.14T + 61T^{2} \)
67 \( 1 - 1.00iT - 67T^{2} \)
71 \( 1 - 6.45T + 71T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 - 1.19T + 79T^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 + 7.29T + 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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

−11.55600511943335852781434535284, −10.71903331077717451307441586625, −10.25989179603639685955839769232, −9.426149915438756562736027514646, −8.068916918924450628884720247867, −7.42567522658600133985910479644, −6.27044265482744543734007670000, −4.84058939518166643161312483564, −4.15397684502326706821031253525, −3.28305222794189869118810119477, 0.42785923145299655776089429301, 2.12798534705374300661897601621, 3.03447413166692801916792642754, 5.15036458068803691678015627947, 5.67229176060720202403518309978, 7.29277915265731209765715638638, 8.178917193068714702186877004839, 8.616853468119460071405087650975, 9.877219962124118192842413110611, 11.27385290597049912506484471065

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