Properties

Label 2-637-91.80-c1-0-16
Degree $2$
Conductor $637$
Sign $-0.692 - 0.721i$
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  + (0.239 + 0.892i)2-s + 1.14i·3-s + (0.993 − 0.573i)4-s + (−1.02 + 3.82i)5-s + (−1.01 + 0.272i)6-s + (2.05 + 2.05i)8-s + 1.69·9-s − 3.65·10-s + (1.48 + 1.48i)11-s + (0.654 + 1.13i)12-s + (3.41 − 1.16i)13-s + (−4.36 − 1.16i)15-s + (−0.195 + 0.339i)16-s + (−1.58 − 2.74i)17-s + (0.405 + 1.51i)18-s + (−0.825 − 0.825i)19-s + ⋯
L(s)  = 1  + (0.169 + 0.630i)2-s + 0.658i·3-s + (0.496 − 0.286i)4-s + (−0.458 + 1.71i)5-s + (−0.415 + 0.111i)6-s + (0.726 + 0.726i)8-s + 0.565·9-s − 1.15·10-s + (0.448 + 0.448i)11-s + (0.188 + 0.327i)12-s + (0.946 − 0.324i)13-s + (−1.12 − 0.301i)15-s + (−0.0489 + 0.0847i)16-s + (−0.384 − 0.666i)17-s + (0.0956 + 0.357i)18-s + (−0.189 − 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.692 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757395 + 1.77577i\)
\(L(\frac12)\) \(\approx\) \(0.757395 + 1.77577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 + 1.16i)T \)
good2 \( 1 + (-0.239 - 0.892i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 1.14iT - 3T^{2} \)
5 \( 1 + (1.02 - 3.82i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.48 - 1.48i)T + 11iT^{2} \)
17 \( 1 + (1.58 + 2.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.825 + 0.825i)T + 19iT^{2} \)
23 \( 1 + (3.26 + 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.584 + 1.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.88 - 1.30i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.26 - 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.85 + 6.93i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.91 + 1.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.2 - 3.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.236 + 0.0633i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 12.3iT - 61T^{2} \)
67 \( 1 + (-7.28 + 7.28i)T - 67iT^{2} \)
71 \( 1 + (-1.60 - 5.98i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.27 + 4.75i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.34 - 2.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.31 - 3.31i)T + 83iT^{2} \)
89 \( 1 + (1.80 + 6.71i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.8 - 4.24i)T + (84.0 - 48.5i)T^{2} \)
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

−10.77234195185918249967387656865, −10.34851978314955791819673214739, −9.300006562976518558467897794745, −7.953030551974324462572060279177, −7.07664902735213848718735897685, −6.66857196306198822783753136413, −5.65159737115681892885625526690, −4.34240148713942887225203032264, −3.42769399572060509764662404617, −2.11995592900108243326488049008, 1.11595837705098455961565943527, 1.86625610907129185265944568725, 3.78306385478012925291356309369, 4.25592784473912282375132977873, 5.71861505692724810342892338155, 6.74042344409192379738655848279, 7.76654912110778942761236197364, 8.449290537972149798984661392978, 9.280972542298213335074529755234, 10.44472097987894087926176109797

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