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Properties

Label 2-63-7.2-c5-0-2
Degree $2$
Conductor $63$
Sign $-0.631 - 0.775i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16 + 27.7i)4-s + (−105.5 + 75.3i)7-s − 427·13-s + (−511. + 886. i)16-s + (−1.57e3 + 2.72e3i)19-s + (1.56e3 + 2.70e3i)25-s + (−3.77e3 − 1.71e3i)28-s + (−1.36e3 − 2.35e3i)31-s + (3.33e3 − 5.76e3i)37-s + 2.24e4·43-s + (5.45e3 − 1.58e4i)49-s + (−6.83e3 − 1.18e4i)52-s + (1.93e4 − 3.34e4i)61-s − 3.27e4·64-s + (1.89e4 + 3.28e4i)67-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.813 + 0.581i)7-s − 0.700·13-s + (−0.499 + 0.866i)16-s + (−0.998 + 1.72i)19-s + (0.5 + 0.866i)25-s + (−0.910 − 0.414i)28-s + (−0.254 − 0.440i)31-s + (0.399 − 0.692i)37-s + 1.85·43-s + (0.324 − 0.945i)49-s + (−0.350 − 0.606i)52-s + (0.664 − 1.15i)61-s − 0.999·64-s + (0.516 + 0.894i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.521762 + 1.09794i\)
\(L(\frac12)\) \(\approx\) \(0.521762 + 1.09794i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (105.5 - 75.3i)T \)
good2 \( 1 + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 427T + 3.71e5T^{2} \)
17 \( 1 + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.57e3 - 2.72e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 2.05e7T^{2} \)
31 \( 1 + (1.36e3 + 2.35e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-3.33e3 + 5.76e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.15e8T^{2} \)
43 \( 1 - 2.24e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.93e4 + 3.34e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.89e4 - 3.28e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 + (-3.90e4 - 6.76e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.54e4 - 7.86e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.34e5T + 8.58e9T^{2} \)
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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

−14.46847659039461493345903431197, −12.80087592266306504327133815938, −12.41851196854919668044258481653, −11.10283308403451755005909291256, −9.709342263118265056967428636651, −8.416888688048899633377404508713, −7.17781922594811372373820198654, −5.88768946107084486704374393855, −3.81877294048210270237322961338, −2.37226808269346661556830159644, 0.53287911063823539317724046957, 2.58941947995594012126329059081, 4.66420969214935304481468916863, 6.29376681186068698486278281416, 7.20800870100179787253572584582, 9.109847853607771610155324182121, 10.21195500788840378869020378486, 11.07485335440873072008018116880, 12.49317080925063326228470202538, 13.64398955816009963221871083143

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