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Properties

Label 2-63-63.38-c3-0-17
Degree $2$
Conductor $63$
Sign $-0.0611 + 0.998i$
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  − 3.06i·2-s + (4.98 − 1.45i)3-s − 1.37·4-s + (2.75 − 4.76i)5-s + (−4.44 − 15.2i)6-s + (8.35 + 16.5i)7-s − 20.2i·8-s + (22.7 − 14.4i)9-s + (−14.5 − 8.42i)10-s + (−51.4 + 29.7i)11-s + (−6.84 + 1.98i)12-s + (−12.7 + 7.35i)13-s + (50.6 − 25.5i)14-s + (6.81 − 27.7i)15-s − 73.0·16-s + (29.3 − 50.7i)17-s + ⋯
L(s)  = 1  − 1.08i·2-s + (0.960 − 0.279i)3-s − 0.171·4-s + (0.246 − 0.426i)5-s + (−0.302 − 1.03i)6-s + (0.451 + 0.892i)7-s − 0.896i·8-s + (0.844 − 0.536i)9-s + (−0.461 − 0.266i)10-s + (−1.41 + 0.814i)11-s + (−0.164 + 0.0478i)12-s + (−0.271 + 0.157i)13-s + (0.965 − 0.488i)14-s + (0.117 − 0.477i)15-s − 1.14·16-s + (0.418 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0611 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0611 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.0611 + 0.998i$
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: \((2,\ 63,\ (\ :3/2),\ -0.0611 + 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.41148 - 1.50059i\)
\(L(\frac12)\) \(\approx\) \(1.41148 - 1.50059i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.98 + 1.45i)T \)
7 \( 1 + (-8.35 - 16.5i)T \)
good2 \( 1 + 3.06iT - 8T^{2} \)
5 \( 1 + (-2.75 + 4.76i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (51.4 - 29.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (12.7 - 7.35i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-29.3 + 50.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (66.1 - 38.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-22.5 - 13.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-217. - 125. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 + (97.2 + 168. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (37.6 + 65.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-210. + 365. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 383.T + 1.03e5T^{2} \)
53 \( 1 + (92.7 + 53.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 533.T + 2.05e5T^{2} \)
61 \( 1 + 41.2iT - 2.26e5T^{2} \)
67 \( 1 - 333.T + 3.00e5T^{2} \)
71 \( 1 + 500. iT - 3.57e5T^{2} \)
73 \( 1 + (-351. - 203. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 150.T + 4.93e5T^{2} \)
83 \( 1 + (493. - 854. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (311. + 539. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-407. - 235. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.96319340005210385032770929647, −12.55916463947635270277774279403, −12.41017306712948269167858339877, −10.68809856743680588400649630101, −9.632883878332185154295369160917, −8.526568382865327535035778426787, −7.13846225150975686931238278759, −4.94542139161836132856205762736, −2.92564337657234364817995247568, −1.80689874576429282520513728820, 2.67817903553017150440575748990, 4.72135909382779573767303371694, 6.40237547572739511909192027890, 7.80645284781129125703157187624, 8.293983293360710836513062853581, 10.16413508250496053214202113413, 10.98146371401532841567075846998, 13.13198325704323136188115596783, 14.00343717877805770779147991473, 14.81374885758633055499246726433

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