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Properties

Label 2-637-13.4-c1-0-0
Degree $2$
Conductor $637$
Sign $-0.878 + 0.477i$
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  + (0.156 + 0.0904i)2-s + (−0.913 + 1.58i)3-s + (−0.983 − 1.70i)4-s + 2.68i·5-s + (−0.285 + 0.165i)6-s − 0.717i·8-s + (−0.167 − 0.289i)9-s + (−0.242 + 0.420i)10-s + (−2.33 − 1.34i)11-s + 3.59·12-s + (−1.92 + 3.05i)13-s + (−4.24 − 2.45i)15-s + (−1.90 + 3.29i)16-s + (−2.38 − 4.12i)17-s − 0.0604i·18-s + (0.163 − 0.0942i)19-s + ⋯
L(s)  = 1  + (0.110 + 0.0639i)2-s + (−0.527 + 0.913i)3-s + (−0.491 − 0.851i)4-s + 1.20i·5-s + (−0.116 + 0.0673i)6-s − 0.253i·8-s + (−0.0557 − 0.0965i)9-s + (−0.0768 + 0.133i)10-s + (−0.703 − 0.406i)11-s + 1.03·12-s + (−0.532 + 0.846i)13-s + (−1.09 − 0.633i)15-s + (−0.475 + 0.823i)16-s + (−0.577 − 1.00i)17-s − 0.0142i·18-s + (0.0374 − 0.0216i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.878 + 0.477i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.878 + 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0580963 - 0.228498i\)
\(L(\frac12)\) \(\approx\) \(0.0580963 - 0.228498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 - 3.05i)T \)
good2 \( 1 + (-0.156 - 0.0904i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.913 - 1.58i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.68iT - 5T^{2} \)
11 \( 1 + (2.33 + 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.38 + 4.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.163 + 0.0942i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 + 3.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.69iT - 31T^{2} \)
37 \( 1 + (6.88 + 3.97i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.70 + 2.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.00 + 6.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.84iT - 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + (6.57 - 3.79i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.205 + 0.356i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.87 - 5.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.89 + 1.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 - 9.11T + 79T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 + (5.10 + 2.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.390 - 0.225i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.89506436712217761711526116774, −10.35041147485704857897104691728, −9.603846661096321957232397451955, −8.788578912317430935024127585053, −7.25049152785925196000307237024, −6.58680670886456919886036471370, −5.38427259611060513180430405619, −4.85212240480091963632307769811, −3.71771676561533540460723238303, −2.31058297831586180100808012332, 0.12850407696643018076043158683, 1.73534964415087999667675464087, 3.35812467213061406444756666927, 4.67526014708418823543590729070, 5.31233254593452819171348397859, 6.52668524285729020794187063033, 7.66628183671025554314095432495, 8.107020767985789404689856257300, 9.071486151058446615140753359376, 9.968200886882348126389073991495

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