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Properties

Label 2-63-63.58-c3-0-4
Degree $2$
Conductor $63$
Sign $-0.551 - 0.834i$
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  + 0.534·2-s + (1.18 + 5.05i)3-s − 7.71·4-s + (0.696 + 1.20i)5-s + (0.633 + 2.70i)6-s + (−2.10 + 18.4i)7-s − 8.39·8-s + (−24.1 + 11.9i)9-s + (0.372 + 0.644i)10-s + (−9.69 + 16.7i)11-s + (−9.14 − 39.0i)12-s + (4.05 − 7.03i)13-s + (−1.12 + 9.83i)14-s + (−5.27 + 4.95i)15-s + 57.2·16-s + (28.8 + 49.9i)17-s + ⋯
L(s)  = 1  + 0.188·2-s + (0.228 + 0.973i)3-s − 0.964·4-s + (0.0623 + 0.107i)5-s + (0.0430 + 0.183i)6-s + (−0.113 + 0.993i)7-s − 0.371·8-s + (−0.896 + 0.444i)9-s + (0.0117 + 0.0203i)10-s + (−0.265 + 0.460i)11-s + (−0.219 − 0.938i)12-s + (0.0866 − 0.150i)13-s + (−0.0214 + 0.187i)14-s + (−0.0908 + 0.0852i)15-s + 0.894·16-s + (0.411 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.551 - 0.834i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.538925 + 1.00219i\)
\(L(\frac12)\) \(\approx\) \(0.538925 + 1.00219i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 - 5.05i)T \)
7 \( 1 + (2.10 - 18.4i)T \)
good2 \( 1 - 0.534T + 8T^{2} \)
5 \( 1 + (-0.696 - 1.20i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (9.69 - 16.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-4.05 + 7.03i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-28.8 - 49.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-35.7 + 61.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-105. - 182. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (78.1 + 135. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 111.T + 2.97e4T^{2} \)
37 \( 1 + (-26.2 + 45.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (201. - 349. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-89.7 - 155. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 92.8T + 1.03e5T^{2} \)
53 \( 1 + (-214. - 371. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 389.T + 2.05e5T^{2} \)
61 \( 1 + 352.T + 2.26e5T^{2} \)
67 \( 1 - 862.T + 3.00e5T^{2} \)
71 \( 1 - 377.T + 3.57e5T^{2} \)
73 \( 1 + (-183. - 318. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 309.T + 4.93e5T^{2} \)
83 \( 1 + (-110. - 191. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-712. + 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (288. + 500. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.02282561956528917940797344599, −13.84384656896950572140883619840, −12.77241117922541288036217637460, −11.44119824673862886501505658316, −9.955664049483253561613131708184, −9.213536609413462137982786967529, −8.108773113470941316405124635489, −5.76054407493889948047907418428, −4.69678067448717280788693359624, −3.10889830413252516455632264197, 0.77634998503586877480792064969, 3.40362143435943329558658199273, 5.20920125986081238894205104325, 6.84983456227887640068519565628, 8.104756530404600315017451691101, 9.212978455343400482651477622138, 10.67212766382648435849706265525, 12.22439305573964479332037984304, 13.16077664431572577951626155352, 13.90351448714352103658737924310

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