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Properties

Label 2-637-91.88-c1-0-34
Degree $2$
Conductor $637$
Sign $0.794 + 0.606i$
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  + (0.395 + 0.228i)2-s + 2.79·3-s + (−0.895 − 1.55i)4-s + (−0.395 + 0.228i)5-s + (1.10 + 0.637i)6-s − 1.73i·8-s + 4.79·9-s − 0.208·10-s − 3.92i·11-s + (−2.5 − 4.33i)12-s + (3.5 + 0.866i)13-s + (−1.10 + 0.637i)15-s + (−1.39 + 2.41i)16-s + (−1.5 − 2.59i)17-s + (1.89 + 1.09i)18-s − 1.37i·19-s + ⋯
L(s)  = 1  + (0.279 + 0.161i)2-s + 1.61·3-s + (−0.447 − 0.775i)4-s + (−0.176 + 0.102i)5-s + (0.450 + 0.260i)6-s − 0.612i·8-s + 1.59·9-s − 0.0660·10-s − 1.18i·11-s + (−0.721 − 1.24i)12-s + (0.970 + 0.240i)13-s + (−0.285 + 0.164i)15-s + (−0.348 + 0.604i)16-s + (−0.363 − 0.630i)17-s + (0.446 + 0.257i)18-s − 0.314i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.794 + 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44785 - 0.827847i\)
\(L(\frac12)\) \(\approx\) \(2.44785 - 0.827847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 + (-0.395 - 0.228i)T + (1 + 1.73i)T^{2} \)
3 \( 1 - 2.79T + 3T^{2} \)
5 \( 1 + (0.395 - 0.228i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 3.92iT - 11T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 1.37iT - 19T^{2} \)
23 \( 1 + (-0.791 + 1.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.79 + 3.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.29 + 4.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.08 - 5.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.6 - 6.15i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 + 4.47iT - 67T^{2} \)
71 \( 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3 + 1.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.02iT - 83T^{2} \)
89 \( 1 + (-13.9 - 8.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.31 + 3.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.44017461861765996665496738927, −9.107101040122626761058533221953, −9.025257781941353437691833399678, −8.081322575432016554647204163194, −6.97267993423042363951242290168, −6.01571060741293661110360917894, −4.77251483431949154101224655195, −3.72216751480771481919384553930, −2.89304293312412584816532785551, −1.28545071516219709894410587161, 1.98924295628334234159921066381, 3.01559399350518113244723723505, 3.99539365037331035321988627286, 4.57006767846061637399974969783, 6.31027458480504666287269709440, 7.65052726775847318673802253791, 8.059837860544257229439088438684, 8.793526369833645044563456202042, 9.574347955672022408075295844068, 10.45246379883673213094459194427

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