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Properties

Label 2-63-63.25-c3-0-7
Degree $2$
Conductor $63$
Sign $0.700 - 0.714i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.09·2-s + (5.00 + 1.41i)3-s + 17.9·4-s + (−3.42 + 5.92i)5-s + (−25.4 − 7.19i)6-s + (−0.241 − 18.5i)7-s − 50.6·8-s + (23.0 + 14.1i)9-s + (17.4 − 30.1i)10-s + (20.5 + 35.5i)11-s + (89.7 + 25.3i)12-s + (31.8 + 55.2i)13-s + (1.23 + 94.3i)14-s + (−25.4 + 24.7i)15-s + 114.·16-s + (38.0 − 65.8i)17-s + ⋯
L(s)  = 1  − 1.80·2-s + (0.962 + 0.271i)3-s + 2.24·4-s + (−0.305 + 0.529i)5-s + (−1.73 − 0.489i)6-s + (−0.0130 − 0.999i)7-s − 2.24·8-s + (0.852 + 0.523i)9-s + (0.550 − 0.954i)10-s + (0.562 + 0.974i)11-s + (2.15 + 0.610i)12-s + (0.680 + 1.17i)13-s + (0.0234 + 1.80i)14-s + (−0.438 + 0.426i)15-s + 1.79·16-s + (0.542 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.700 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.700 - 0.714i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.700 - 0.714i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.807169 + 0.339012i\)
\(L(\frac12)\) \(\approx\) \(0.807169 + 0.339012i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.00 - 1.41i)T \)
7 \( 1 + (0.241 + 18.5i)T \)
good2 \( 1 + 5.09T + 8T^{2} \)
5 \( 1 + (3.42 - 5.92i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20.5 - 35.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-31.8 - 55.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-38.0 + 65.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46.7 - 81.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (21.0 - 36.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-21.9 + 37.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 200.T + 2.97e4T^{2} \)
37 \( 1 + (52.1 + 90.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (108. + 188. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-236. + 409. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 17.8T + 1.03e5T^{2} \)
53 \( 1 + (-211. + 365. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 291.T + 2.05e5T^{2} \)
61 \( 1 + 154.T + 2.26e5T^{2} \)
67 \( 1 + 838.T + 3.00e5T^{2} \)
71 \( 1 + 940.T + 3.57e5T^{2} \)
73 \( 1 + (-65.8 + 113. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 620.T + 4.93e5T^{2} \)
83 \( 1 + (-102. + 177. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-432. - 749. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-331. + 573. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.80893809827326768907817870576, −13.85240144363301874954472356179, −11.86752190372582382647966389887, −10.67773457024261715842107377074, −9.770186687469928910759546698522, −8.947314006948785413252085394733, −7.47297311290560603970999561905, −7.08141732409914285412513103135, −3.71893228104131027220757583891, −1.67470356833191887468302229357, 1.12197241959487120018474436019, 3.00872075705702476697106215035, 6.17797512043969974230808217003, 7.81944505481407992123651475750, 8.586053419531365800079653903585, 9.164174156331377132840473641902, 10.54877562783033631043680035414, 11.84844165708937765702821450656, 13.00780017057069619084795594791, 14.74955233962544634312042617316

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