Properties

Label 2-637-49.39-c1-0-10
Degree $2$
Conductor $637$
Sign $-0.292 - 0.956i$
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  + (−1.64 − 0.248i)2-s + (0.245 − 0.167i)3-s + (0.746 + 0.230i)4-s + (0.200 + 2.67i)5-s + (−0.445 + 0.214i)6-s + (2.62 + 0.327i)7-s + (1.83 + 0.881i)8-s + (−1.06 + 2.71i)9-s + (0.334 − 4.46i)10-s + (0.884 + 2.25i)11-s + (0.221 − 0.0683i)12-s + (−0.623 + 0.781i)13-s + (−4.24 − 1.19i)14-s + (0.496 + 0.622i)15-s + (−4.09 − 2.78i)16-s + (−1.10 − 1.02i)17-s + ⋯
L(s)  = 1  + (−1.16 − 0.175i)2-s + (0.141 − 0.0964i)3-s + (0.373 + 0.115i)4-s + (0.0897 + 1.19i)5-s + (−0.181 + 0.0876i)6-s + (0.992 + 0.123i)7-s + (0.647 + 0.311i)8-s + (−0.354 + 0.903i)9-s + (0.105 − 1.41i)10-s + (0.266 + 0.679i)11-s + (0.0639 − 0.0197i)12-s + (−0.172 + 0.216i)13-s + (−1.13 − 0.318i)14-s + (0.128 + 0.160i)15-s + (−1.02 − 0.697i)16-s + (−0.266 − 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.425391 + 0.575129i\)
\(L(\frac12)\) \(\approx\) \(0.425391 + 0.575129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 - 0.327i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (1.64 + 0.248i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-0.245 + 0.167i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.200 - 2.67i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-0.884 - 2.25i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (1.10 + 1.02i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (1.62 + 2.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.39 - 3.14i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (0.0150 - 0.0657i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.87 + 3.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.3 - 3.19i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (1.25 + 0.605i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.12 - 1.50i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-10.0 - 1.51i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-1.67 - 0.517i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.534 - 7.13i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (3.00 - 0.925i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.497 + 0.862i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0794 - 0.348i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-3.93 + 0.592i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-3.62 - 6.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.69 + 3.37i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (5.05 - 12.8i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−10.71819352772025034036166436271, −10.06228130928821520139592554301, −9.081002047630653145435733758414, −8.266044574672911699245992491096, −7.48929711411655515344732485652, −6.85298853131443116683540422880, −5.35243802019792157169098562761, −4.35533891300183550944159236941, −2.56943350794062089022919525022, −1.78680761708730235677027216731, 0.58263050946334485411011372018, 1.73978720247128447178933851090, 3.79231939745471748418128081762, 4.73850800648303669688090957082, 5.84302245243788599456835567548, 7.03543761857910093365152901542, 8.293712045693269952991746621238, 8.496712163209967351977568878095, 9.141533766866915690259175317034, 10.14230956306476183114307726325

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