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Properties

Label 2-637-49.43-c1-0-2
Degree $2$
Conductor $637$
Sign $0.00508 - 0.999i$
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  + (0.311 − 0.390i)2-s + (−1.89 − 0.911i)3-s + (0.389 + 1.70i)4-s + (−2.41 − 1.16i)5-s + (−0.945 + 0.455i)6-s + (0.718 − 2.54i)7-s + (1.68 + 0.812i)8-s + (0.883 + 1.10i)9-s + (−1.20 + 0.580i)10-s + (−1.57 + 1.97i)11-s + (0.818 − 3.58i)12-s + (0.623 − 0.781i)13-s + (−0.770 − 1.07i)14-s + (3.51 + 4.40i)15-s + (−2.31 + 1.11i)16-s + (−0.652 + 2.85i)17-s + ⋯
L(s)  = 1  + (0.220 − 0.275i)2-s + (−1.09 − 0.526i)3-s + (0.194 + 0.853i)4-s + (−1.07 − 0.520i)5-s + (−0.385 + 0.185i)6-s + (0.271 − 0.962i)7-s + (0.596 + 0.287i)8-s + (0.294 + 0.369i)9-s + (−0.381 + 0.183i)10-s + (−0.475 + 0.596i)11-s + (0.236 − 1.03i)12-s + (0.172 − 0.216i)13-s + (−0.205 − 0.286i)14-s + (0.906 + 1.13i)15-s + (−0.578 + 0.278i)16-s + (−0.158 + 0.693i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00508 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00508 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.00508 - 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.00508 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298987 + 0.297470i\)
\(L(\frac12)\) \(\approx\) \(0.298987 + 0.297470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.718 + 2.54i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (-0.311 + 0.390i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (1.89 + 0.911i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (2.41 + 1.16i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.57 - 1.97i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (0.652 - 2.85i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 0.653T + 19T^{2} \)
23 \( 1 + (-0.927 - 4.06i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (1.41 - 6.21i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 2.14T + 31T^{2} \)
37 \( 1 + (1.05 - 4.63i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-6.69 - 3.22i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.22 - 1.55i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.15 - 1.45i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.09 - 4.78i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (7.68 - 3.70i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.103 + 0.455i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 9.76T + 67T^{2} \)
71 \( 1 + (1.65 + 7.26i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.88 + 7.37i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 4.47T + 79T^{2} \)
83 \( 1 + (-0.977 - 1.22i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.46 + 1.84i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 8.04T + 97T^{2} \)
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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

−11.13636584933467470582711274946, −10.38833519891134120688526755933, −8.844586167127536886473554537624, −7.78592588454838135170923376568, −7.45571864916975597495390521791, −6.47003073620015391848851141920, −5.08714297152405811493050906103, −4.29466746579912041334518399005, −3.35147589056010535228887512636, −1.42291489016642614722771739137, 0.25437306105440524103751227160, 2.47883719017537529196679342207, 4.07497860931690388279312029017, 5.04442676684436695705100977989, 5.72121917793889200474258385656, 6.51263000073223872382006943527, 7.54966929575239249747887899555, 8.601335514379973171662001923362, 9.721579714465433691797928223762, 10.67549925804035743255997392048

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