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Properties

Label 2-637-91.76-c1-0-19
Degree $2$
Conductor $637$
Sign $-0.219 - 0.975i$
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  + (−0.360 + 1.34i)2-s + (1.25 − 0.724i)3-s + (0.0536 + 0.0309i)4-s + (−0.470 + 0.470i)5-s + (0.521 + 1.94i)6-s + (−2.02 + 2.02i)8-s + (−0.450 + 0.780i)9-s + (−0.463 − 0.802i)10-s + (4.65 + 1.24i)11-s + 0.0896·12-s + (3.60 + 0.0282i)13-s + (−0.249 + 0.931i)15-s + (−1.93 − 3.35i)16-s + (−0.233 + 0.405i)17-s + (−0.887 − 0.887i)18-s + (−0.873 − 3.26i)19-s + ⋯
L(s)  = 1  + (−0.254 + 0.950i)2-s + (0.724 − 0.418i)3-s + (0.0268 + 0.0154i)4-s + (−0.210 + 0.210i)5-s + (0.213 + 0.795i)6-s + (−0.717 + 0.717i)8-s + (−0.150 + 0.260i)9-s + (−0.146 − 0.253i)10-s + (1.40 + 0.376i)11-s + 0.0258·12-s + (0.999 + 0.00782i)13-s + (−0.0644 + 0.240i)15-s + (−0.484 − 0.838i)16-s + (−0.0567 + 0.0982i)17-s + (−0.209 − 0.209i)18-s + (−0.200 − 0.748i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09734 + 1.37111i\)
\(L(\frac12)\) \(\approx\) \(1.09734 + 1.37111i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 0.0282i)T \)
good2 \( 1 + (0.360 - 1.34i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.25 + 0.724i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.470 - 0.470i)T - 5iT^{2} \)
11 \( 1 + (-4.65 - 1.24i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.233 - 0.405i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.873 + 3.26i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.02 - 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.00 + 3.00i)T - 31iT^{2} \)
37 \( 1 + (-3.26 - 0.873i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.68 - 0.986i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.06 - 7.06i)T + 47iT^{2} \)
53 \( 1 + 4.40T + 53T^{2} \)
59 \( 1 + (5.91 - 1.58i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.21 + 2.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.42 + 9.03i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.19 + 0.857i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.0824 + 0.0824i)T + 73iT^{2} \)
79 \( 1 + 0.389T + 79T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (-2.50 + 9.34i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.61 + 17.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.05019546649514068719500019031, −9.442773487696822744128723035246, −8.940383337521430248047040358535, −7.939318559639960748181542498086, −7.50882876410877843970169980103, −6.48011782450346136360460352689, −5.84775975380626039569180639768, −4.25284265499343326293398889986, −3.09057530520186516992617052675, −1.80749676438357235079802154719, 1.03662276961011771979585820988, 2.41176374717836027249070879198, 3.68339055255828324503300295776, 4.03566795782152115439987757974, 6.00386979030488113963896660097, 6.57317476249489446103948404625, 8.167436268444612787467893648982, 8.845779188614438389361338988900, 9.458105187751152261991164843165, 10.35617573831781296484005999002

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