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Properties

Label 2-637-91.19-c1-0-18
Degree $2$
Conductor $637$
Sign $-0.746 - 0.664i$
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  + (2.26 + 0.607i)2-s + 2.76i·3-s + (3.03 + 1.75i)4-s + (−1.53 + 0.412i)5-s + (−1.67 + 6.25i)6-s + (2.50 + 2.50i)8-s − 4.61·9-s − 3.74·10-s + (2.22 + 2.22i)11-s + (−4.84 + 8.39i)12-s + (−1.04 − 3.44i)13-s + (−1.13 − 4.24i)15-s + (0.649 + 1.12i)16-s + (0.320 − 0.555i)17-s + (−10.4 − 2.80i)18-s + (5.57 + 5.57i)19-s + ⋯
L(s)  = 1  + (1.60 + 0.429i)2-s + 1.59i·3-s + (1.51 + 0.877i)4-s + (−0.688 + 0.184i)5-s + (−0.684 + 2.55i)6-s + (0.886 + 0.886i)8-s − 1.53·9-s − 1.18·10-s + (0.669 + 0.669i)11-s + (−1.39 + 2.42i)12-s + (−0.290 − 0.956i)13-s + (−0.293 − 1.09i)15-s + (0.162 + 0.281i)16-s + (0.0778 − 0.134i)17-s + (−2.46 − 0.661i)18-s + (1.27 + 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.746 - 0.664i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.746 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13177 + 2.97327i\)
\(L(\frac12)\) \(\approx\) \(1.13177 + 2.97327i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 + 3.44i)T \)
good2 \( 1 + (-2.26 - 0.607i)T + (1.73 + i)T^{2} \)
3 \( 1 - 2.76iT - 3T^{2} \)
5 \( 1 + (1.53 - 0.412i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.22 - 2.22i)T + 11iT^{2} \)
17 \( 1 + (-0.320 + 0.555i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.57 - 5.57i)T + 19iT^{2} \)
23 \( 1 + (-0.126 + 0.0730i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.49 + 2.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 + 6.46i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.00 - 3.75i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.60 + 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.42 + 1.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.816 + 3.04i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.66 - 6.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.07 - 4.00i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 4.50iT - 61T^{2} \)
67 \( 1 + (1.00 - 1.00i)T - 67iT^{2} \)
71 \( 1 + (-13.8 - 3.70i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.81 + 1.82i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.316 + 0.548i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.07 - 1.07i)T + 83iT^{2} \)
89 \( 1 + (13.1 + 3.51i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.0487 + 0.181i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.14529142130777809732813658522, −10.04651841563740129406306110368, −9.494374002611885236398232835091, −8.030198137086876845063994782534, −7.24711142006280651826777208997, −5.90822004153957133973787710304, −5.27297317098304308335463832053, −4.25928751912706968872957332167, −3.79735621020833783932251844853, −2.88955038399279905358383629840, 1.15314887709731980736404198341, 2.45501622786377158603497832777, 3.49258646123548889575679978656, 4.60887363060325820979277260342, 5.66870285805522293105557676402, 6.65617440276749868178514865562, 7.13350847830683484363097950806, 8.245690195898548974958273092608, 9.255515050443338080883170645725, 10.97562232054596410720540643725

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