↑

Properties

Label 2-637-91.83-c1-0-14
Degree $2$
Conductor $637$
Sign $0.473 - 0.880i$
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  + (0.546 − 0.546i)2-s + 0.487i·3-s + 1.40i·4-s + (1.29 + 1.29i)5-s + (0.266 + 0.266i)6-s + (1.85 + 1.85i)8-s + 2.76·9-s + 1.41·10-s + (−0.725 − 0.725i)11-s − 0.683·12-s + (−0.266 + 3.59i)13-s + (−0.628 + 0.628i)15-s − 0.773·16-s − 5.20·17-s + (1.50 − 1.50i)18-s + (−3.71 − 3.71i)19-s + ⋯
L(s)  = 1  + (0.386 − 0.386i)2-s + 0.281i·3-s + 0.701i·4-s + (0.577 + 0.577i)5-s + (0.108 + 0.108i)6-s + (0.657 + 0.657i)8-s + 0.920·9-s + 0.445·10-s + (−0.218 − 0.218i)11-s − 0.197·12-s + (−0.0738 + 0.997i)13-s + (−0.162 + 0.162i)15-s − 0.193·16-s − 1.26·17-s + (0.355 − 0.355i)18-s + (−0.852 − 0.852i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.473 - 0.880i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75031 + 1.04582i\)
\(L(\frac12)\) \(\approx\) \(1.75031 + 1.04582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.266 - 3.59i)T \)
good2 \( 1 + (-0.546 + 0.546i)T - 2iT^{2} \)
3 \( 1 - 0.487iT - 3T^{2} \)
5 \( 1 + (-1.29 - 1.29i)T + 5iT^{2} \)
11 \( 1 + (0.725 + 0.725i)T + 11iT^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 + (3.71 + 3.71i)T + 19iT^{2} \)
23 \( 1 - 0.843iT - 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + (-4.16 - 4.16i)T + 31iT^{2} \)
37 \( 1 + (-4.41 - 4.41i)T + 37iT^{2} \)
41 \( 1 + (0.0927 + 0.0927i)T + 41iT^{2} \)
43 \( 1 + 7.36iT - 43T^{2} \)
47 \( 1 + (1.59 - 1.59i)T - 47iT^{2} \)
53 \( 1 + 6.77T + 53T^{2} \)
59 \( 1 + (-7.12 + 7.12i)T - 59iT^{2} \)
61 \( 1 - 1.30iT - 61T^{2} \)
67 \( 1 + (-3.04 + 3.04i)T - 67iT^{2} \)
71 \( 1 + (6.02 - 6.02i)T - 71iT^{2} \)
73 \( 1 + (-8.02 + 8.02i)T - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (4.16 + 4.16i)T + 83iT^{2} \)
89 \( 1 + (-5.48 + 5.48i)T - 89iT^{2} \)
97 \( 1 + (-2.49 - 2.49i)T + 97iT^{2} \)
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

−10.75811779778568712193954492185, −10.08097302455907416601606306488, −9.004660479127969225547715792991, −8.231603867519221887978993619368, −6.87460042974464270057236188575, −6.55101472154905794327488016224, −4.74687487832657431318160321589, −4.30913401159770510682097133274, −2.93526063871829059700012341682, −2.02828627254681425335286164149, 1.06445673402800535356222200329, 2.29794541273832608792146566489, 4.25081006623047643108265070592, 4.90672916439851463697088340978, 5.99768789736344247872786201774, 6.59859959109181701999531694922, 7.67796525284293446618570288300, 8.650731785170831853362108544669, 9.820416634265258778655388322329, 10.19486620281165856679156132796

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