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Properties

Label 2-637-91.74-c1-0-16
Degree $2$
Conductor $637$
Sign $0.996 - 0.0856i$
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  − 2.74·2-s + (0.682 + 1.18i)3-s + 5.51·4-s + (0.370 + 0.641i)5-s + (−1.87 − 3.23i)6-s − 9.63·8-s + (0.568 − 0.984i)9-s + (−1.01 − 1.75i)10-s + (0.682 + 1.18i)11-s + (3.76 + 6.51i)12-s + (−0.301 − 3.59i)13-s + (−0.505 + 0.875i)15-s + 15.3·16-s + 4.14·17-s + (−1.55 + 2.69i)18-s + (3.63 − 6.29i)19-s + ⋯
L(s)  = 1  − 1.93·2-s + (0.393 + 0.682i)3-s + 2.75·4-s + (0.165 + 0.287i)5-s + (−0.763 − 1.32i)6-s − 3.40·8-s + (0.189 − 0.328i)9-s + (−0.321 − 0.556i)10-s + (0.205 + 0.356i)11-s + (1.08 + 1.88i)12-s + (−0.0837 − 0.996i)13-s + (−0.130 + 0.226i)15-s + 3.84·16-s + 1.00·17-s + (−0.367 + 0.636i)18-s + (0.833 − 1.44i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.996 - 0.0856i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.996 - 0.0856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.784391 + 0.0336391i\)
\(L(\frac12)\) \(\approx\) \(0.784391 + 0.0336391i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.301 + 3.59i)T \)
good2 \( 1 + 2.74T + 2T^{2} \)
3 \( 1 + (-0.682 - 1.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.370 - 0.641i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.682 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 + (-3.63 + 6.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 + (-0.203 + 0.353i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.38 - 2.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.10T + 37T^{2} \)
41 \( 1 + (-0.627 + 1.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.870 - 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.92 + 5.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.28 - 3.95i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + (-3.26 + 5.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.87 - 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.40 - 4.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.03 + 5.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.56 - 7.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 1.76T + 89T^{2} \)
97 \( 1 + (-4.76 - 8.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08211140444523328674326077516, −9.876705545859734866322305272872, −8.960160946340526045049502777568, −8.257202568425130852334045708318, −7.27935837038036312426955208581, −6.61751254815086221209088914382, −5.32477010593727015746022517965, −3.46063082662948512982250838881, −2.54943545075562989168348332727, −0.908313246963195820354831000611, 1.24531006439571584210548793155, 1.99210851853087117234201617422, 3.39635534643680282835498037498, 5.53502929125539901718185676385, 6.59776774921474842778788077812, 7.42996620298992785939675402700, 8.013612307421042402880434264037, 8.766425729732371279991381884654, 9.628667489488321025323223765277, 10.20682398317165800473484832384

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