↑

Properties

Label 2-637-91.81-c1-0-2
Degree $2$
Conductor $637$
Sign $-0.270 - 0.962i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + (0.651 − 1.12i)2-s − 2.88·3-s + (0.151 + 0.262i)4-s + (1.44 + 2.49i)5-s + (−1.87 + 3.25i)6-s + 3·8-s + 5.30·9-s + 3.75·10-s − 5.90·11-s + (−0.436 − 0.755i)12-s + (−3.31 + 1.41i)13-s + (−4.15 − 7.19i)15-s + (1.65 − 2.86i)16-s + (−0.436 − 0.755i)17-s + (3.45 − 5.98i)18-s − 2.88·19-s + ⋯
L(s)  = 1  + (0.460 − 0.797i)2-s − 1.66·3-s + (0.0756 + 0.131i)4-s + (0.644 + 1.11i)5-s + (−0.766 + 1.32i)6-s + 1.06·8-s + 1.76·9-s + 1.18·10-s − 1.78·11-s + (−0.125 − 0.218i)12-s + (−0.920 + 0.391i)13-s + (−1.07 − 1.85i)15-s + (0.412 − 0.715i)16-s + (−0.105 − 0.183i)17-s + (0.814 − 1.41i)18-s − 0.661·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.270 - 0.962i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.270 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401493 + 0.529721i\)
\(L(\frac12)\) \(\approx\) \(0.401493 + 0.529721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (3.31 - 1.41i)T \)
good2 \( 1 + (-0.651 + 1.12i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 + (-1.44 - 2.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
17 \( 1 + (0.436 + 0.755i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.88T + 19T^{2} \)
23 \( 1 + (3.30 - 5.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.651 + 1.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.436 + 0.755i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.697 - 1.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.19 + 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.80 - 8.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.31 - 5.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 5.76T + 61T^{2} \)
67 \( 1 - T + 67T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.88 - 4.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.302 + 0.524i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + (-4.32 + 7.48i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.88 - 6.73i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.98561458214860426518022685502, −10.24579739677536565264119769932, −9.912465429173331996474945934118, −7.82688317178886329083062210592, −7.11530423119909990739003302320, −6.19711589203445344710549032243, −5.29186700700751361976984848614, −4.48289715320759449117064495917, −2.96290337409701181636073737890, −2.00406772008254332923907426509, 0.35232875820824479073730201156, 2.02734045614282832183783224516, 4.65434739539467427610017102900, 4.99016298812799822771600940533, 5.68690271964038464712687686915, 6.34428015578737401457285997680, 7.39689825945529928656404350941, 8.309639699189783011609097856314, 9.794535540601185319282320462248, 10.46473694669118604717579500192

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