Properties

Label 2-637-91.80-c1-0-7
Degree $2$
Conductor $637$
Sign $0.999 + 0.0283i$
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  + (−0.433 − 1.61i)2-s + 0.637i·3-s + (−0.700 + 0.404i)4-s + (−0.520 + 1.94i)5-s + (1.03 − 0.276i)6-s + (−1.41 − 1.41i)8-s + 2.59·9-s + 3.36·10-s + (0.694 + 0.694i)11-s + (−0.258 − 0.446i)12-s + (−1.60 + 3.22i)13-s + (−1.23 − 0.331i)15-s + (−2.48 + 4.29i)16-s + (2.99 + 5.18i)17-s + (−1.12 − 4.19i)18-s + (−1.98 − 1.98i)19-s + ⋯
L(s)  = 1  + (−0.306 − 1.14i)2-s + 0.368i·3-s + (−0.350 + 0.202i)4-s + (−0.232 + 0.868i)5-s + (0.421 − 0.112i)6-s + (−0.498 − 0.498i)8-s + 0.864·9-s + 1.06·10-s + (0.209 + 0.209i)11-s + (−0.0744 − 0.129i)12-s + (−0.446 + 0.894i)13-s + (−0.319 − 0.0856i)15-s + (−0.620 + 1.07i)16-s + (0.725 + 1.25i)17-s + (−0.265 − 0.989i)18-s + (−0.455 − 0.455i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.999 + 0.0283i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.999 + 0.0283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24894 - 0.0177263i\)
\(L(\frac12)\) \(\approx\) \(1.24894 - 0.0177263i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 3.22i)T \)
good2 \( 1 + (0.433 + 1.61i)T + (-1.73 + i)T^{2} \)
3 \( 1 - 0.637iT - 3T^{2} \)
5 \( 1 + (0.520 - 1.94i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.694 - 0.694i)T + 11iT^{2} \)
17 \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.98 + 1.98i)T + 19iT^{2} \)
23 \( 1 + (2.58 + 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.65 - 6.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.34 + 2.23i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.63 + 1.24i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.886 - 3.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.748 + 0.432i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.96 - 0.794i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.16 - 5.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.491 - 0.131i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + (-0.606 + 0.606i)T - 67iT^{2} \)
71 \( 1 + (3.01 + 11.2i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.377 + 1.40i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (-2.07 - 7.75i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.0 - 3.23i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.62487351470044173883296078027, −9.939172868130053763671352665193, −9.294599745243796929478903174463, −8.111906627487017367537528155598, −6.89485398990080517623267345105, −6.33968674733554735516321263912, −4.58399450332794644197569427999, −3.75833017010978039966705807725, −2.69086883752063381119208405265, −1.48950603259582331764251535554, 0.826449252991981878862121558025, 2.70557784602819727034742259900, 4.36790854297317510898493312971, 5.29895417051102113277102658376, 6.24613683269525558713419371163, 7.17409310745581790844871526175, 7.934525076086748730151396770076, 8.429541404930903156003435503975, 9.554892024030690425935484387245, 10.25013717677856765301916943156

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