↑

Properties

Label 2-637-91.4-c1-0-25
Degree $2$
Conductor $637$
Sign $0.379 + 0.925i$
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  − 1.04i·2-s + (0.384 + 0.666i)3-s + 0.918·4-s + (−1.45 + 0.838i)5-s + (0.692 − 0.400i)6-s − 3.03i·8-s + (1.20 − 2.08i)9-s + (0.871 + 1.51i)10-s + (0.465 − 0.268i)11-s + (0.353 + 0.611i)12-s + (−1.96 − 3.02i)13-s + (−1.11 − 0.644i)15-s − 1.32·16-s + 5.62·17-s + (−2.16 − 1.25i)18-s + (−1.74 − 1.00i)19-s + ⋯
L(s)  = 1  − 0.735i·2-s + (0.222 + 0.384i)3-s + 0.459·4-s + (−0.649 + 0.374i)5-s + (0.282 − 0.163i)6-s − 1.07i·8-s + (0.401 − 0.695i)9-s + (0.275 + 0.477i)10-s + (0.140 − 0.0810i)11-s + (0.101 + 0.176i)12-s + (−0.544 − 0.838i)13-s + (−0.288 − 0.166i)15-s − 0.330·16-s + 1.36·17-s + (−0.511 − 0.295i)18-s + (−0.400 − 0.231i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45686 - 0.976931i\)
\(L(\frac12)\) \(\approx\) \(1.45686 - 0.976931i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.96 + 3.02i)T \)
good2 \( 1 + 1.04iT - 2T^{2} \)
3 \( 1 + (-0.384 - 0.666i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.45 - 0.838i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.465 + 0.268i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.62T + 17T^{2} \)
19 \( 1 + (1.74 + 1.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.67T + 23T^{2} \)
29 \( 1 + (-2.43 + 4.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.28 - 1.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.38iT - 37T^{2} \)
41 \( 1 + (1.07 + 0.620i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.63 + 2.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.29 + 1.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.73 - 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.07iT - 59T^{2} \)
61 \( 1 + (-1.88 + 3.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.31 - 4.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 0.760i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.5 + 7.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.13 + 7.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.42iT - 83T^{2} \)
89 \( 1 - 2.13iT - 89T^{2} \)
97 \( 1 + (5.58 - 3.22i)T + (48.5 - 84.0i)T^{2} \)
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

−10.37806197960546538682634351944, −9.895539276361343480506495846120, −8.877037006201941822487402640818, −7.64663330353711012032606314503, −7.04104800451516447656547053942, −5.92922414889165370305295961359, −4.51463215488375549971448387468, −3.40861949876592949671877243629, −2.87542950943212090584481318186, −1.04825837401322489914484482723, 1.59553187679656977186463008229, 2.92309714112094775590921562999, 4.46194295166321880364123735142, 5.30235707880439051456310312668, 6.55617003713404636582483554785, 7.25587840646423694672319651898, 7.943769247322219577146893111075, 8.607497235705547321849319029548, 9.826831585612807698322160189192, 10.80571800454086232420701412585

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