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Properties

Label 2-63-63.38-c3-0-14
Degree $2$
Conductor $63$
Sign $0.0938 + 0.995i$
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  − 3.72i·2-s + (3.89 + 3.43i)3-s − 5.84·4-s + (1.33 − 2.30i)5-s + (12.7 − 14.4i)6-s + (8.98 − 16.1i)7-s − 8.01i·8-s + (3.35 + 26.7i)9-s + (−8.57 − 4.94i)10-s + (58.8 − 33.9i)11-s + (−22.7 − 20.0i)12-s + (−69.6 + 40.2i)13-s + (−60.2 − 33.4i)14-s + (13.1 − 4.40i)15-s − 76.5·16-s + (−29.6 + 51.2i)17-s + ⋯
L(s)  = 1  − 1.31i·2-s + (0.749 + 0.661i)3-s − 0.730·4-s + (0.118 − 0.206i)5-s + (0.870 − 0.986i)6-s + (0.485 − 0.874i)7-s − 0.354i·8-s + (0.124 + 0.992i)9-s + (−0.271 − 0.156i)10-s + (1.61 − 0.931i)11-s + (−0.547 − 0.483i)12-s + (−1.48 + 0.858i)13-s + (−1.15 − 0.638i)14-s + (0.225 − 0.0757i)15-s − 1.19·16-s + (−0.422 + 0.731i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0938 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0938 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.0938 + 0.995i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.0938 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.40590 - 1.27966i\)
\(L(\frac12)\) \(\approx\) \(1.40590 - 1.27966i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.89 - 3.43i)T \)
7 \( 1 + (-8.98 + 16.1i)T \)
good2 \( 1 + 3.72iT - 8T^{2} \)
5 \( 1 + (-1.33 + 2.30i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-58.8 + 33.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (69.6 - 40.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (29.6 - 51.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (5.86 - 3.38i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-109. - 63.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (114. + 66.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 72.5iT - 2.97e4T^{2} \)
37 \( 1 + (63.8 + 110. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (4.93 + 8.54i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (108. - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 185.T + 1.03e5T^{2} \)
53 \( 1 + (-174. - 100. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 178.T + 2.05e5T^{2} \)
61 \( 1 - 225. iT - 2.26e5T^{2} \)
67 \( 1 - 438.T + 3.00e5T^{2} \)
71 \( 1 + 533. iT - 3.57e5T^{2} \)
73 \( 1 + (-287. - 165. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (-590. + 1.02e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (66.3 + 114. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-761. - 439. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.09480930764884432655744573544, −13.08123006379119926894064665585, −11.63336531440501645429102981913, −10.88129363241858095027837932739, −9.657551730178586074135243042767, −8.901930140357458165620077295022, −7.09697443439282307048096979688, −4.53879446689197646532343880183, −3.49568466737368440689713723347, −1.63975045480217539060718318754, 2.36615973943433889617429565867, 4.93889728203859283181406911665, 6.58317285256150377844744325907, 7.33377490979516154727118268864, 8.578783962303789070057673731396, 9.506453065482974140002694761465, 11.70623740756408160586817003466, 12.65773938094018401965965451142, 14.23716561161949248690823253024, 14.78931765321051389930394911497

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