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Properties

Label 2-637-91.4-c1-0-28
Degree $2$
Conductor $637$
Sign $0.721 + 0.692i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.120i·2-s + (−0.291 − 0.504i)3-s + 1.98·4-s + (1.46 − 0.844i)5-s + (0.0606 − 0.0350i)6-s + 0.479i·8-s + (1.33 − 2.30i)9-s + (0.101 + 0.175i)10-s + (0.315 − 0.182i)11-s + (−0.578 − 1.00i)12-s + (−1.80 − 3.12i)13-s + (−0.851 − 0.491i)15-s + 3.91·16-s − 3.18·17-s + (0.277 + 0.160i)18-s + (1.25 + 0.721i)19-s + ⋯
L(s)  = 1  + 0.0851i·2-s + (−0.168 − 0.291i)3-s + 0.992·4-s + (0.653 − 0.377i)5-s + (0.0247 − 0.0143i)6-s + 0.169i·8-s + (0.443 − 0.768i)9-s + (0.0321 + 0.0556i)10-s + (0.0952 − 0.0549i)11-s + (−0.166 − 0.289i)12-s + (−0.499 − 0.866i)13-s + (−0.219 − 0.126i)15-s + 0.978·16-s − 0.772·17-s + (0.0653 + 0.0377i)18-s + (0.286 + 0.165i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.721 + 0.692i$
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: \((2,\ 637,\ (\ :1/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83748 - 0.738999i\)
\(L(\frac12)\) \(\approx\) \(1.83748 - 0.738999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 3.12i)T \)
good2 \( 1 - 0.120iT - 2T^{2} \)
3 \( 1 + (0.291 + 0.504i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.46 + 0.844i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.315 + 0.182i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 + (-1.25 - 0.721i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.08T + 23T^{2} \)
29 \( 1 + (4.09 - 7.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.06 + 2.34i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.31iT - 37T^{2} \)
41 \( 1 + (-5.04 - 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.386 + 0.669i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.0 + 6.39i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.685 - 1.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.36iT - 59T^{2} \)
61 \( 1 + (4.51 - 7.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.6 + 6.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.13 - 3.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.87 - 1.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.44 - 5.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.567iT - 83T^{2} \)
89 \( 1 + 1.13iT - 89T^{2} \)
97 \( 1 + (6.86 - 3.96i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.60910371834135039190863680706, −9.563010317425095577777062403692, −8.879783431941949850580336577072, −7.46951108740273431941886103998, −7.03885634414560308994693533964, −5.93127205291845006847643982694, −5.31203048137325956897077835322, −3.71147652751609942821916987082, −2.44056111630018915532300873596, −1.20651338731380130346349792496, 1.83170630617022826383158577384, 2.63531714927308126622399491379, 4.15278103686340291585497677894, 5.26255519082608081303937681586, 6.31502138180903185824786888120, 7.03865343656464803379684963313, 7.83599589576829564022957789500, 9.245905702928346193104911239151, 9.906753134434608642558479335373, 10.83724016187034656023323367020

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