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Properties

Label 2-63-9.2-c2-0-0
Degree $2$
Conductor $63$
Sign $-0.999 + 0.0134i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.296 + 0.170i)2-s + (−2.98 + 0.281i)3-s + (−1.94 − 3.36i)4-s + (−7.71 + 4.45i)5-s + (−0.932 − 0.427i)6-s + (−1.32 + 2.29i)7-s − 2.69i·8-s + (8.84 − 1.68i)9-s − 3.04·10-s + (0.233 + 0.134i)11-s + (6.74 + 9.49i)12-s + (−9.41 − 16.3i)13-s + (−0.783 + 0.452i)14-s + (21.7 − 15.4i)15-s + (−7.30 + 12.6i)16-s + 11.7i·17-s + ⋯
L(s)  = 1  + (0.148 + 0.0854i)2-s + (−0.995 + 0.0938i)3-s + (−0.485 − 0.840i)4-s + (−1.54 + 0.890i)5-s + (−0.155 − 0.0712i)6-s + (−0.188 + 0.327i)7-s − 0.336i·8-s + (0.982 − 0.186i)9-s − 0.304·10-s + (0.0211 + 0.0122i)11-s + (0.562 + 0.791i)12-s + (−0.724 − 1.25i)13-s + (−0.0559 + 0.0323i)14-s + (1.45 − 1.03i)15-s + (−0.456 + 0.790i)16-s + 0.692i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0134i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.999 + 0.0134i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.999 + 0.0134i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(9.83085\times10^{-5} - 0.0146720i\)
\(L(\frac12)\) \(\approx\) \(9.83085\times10^{-5} - 0.0146720i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.98 - 0.281i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good2 \( 1 + (-0.296 - 0.170i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (7.71 - 4.45i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-0.233 - 0.134i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.41 + 16.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 11.7iT - 289T^{2} \)
19 \( 1 + 0.353T + 361T^{2} \)
23 \( 1 + (25.5 - 14.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (4.73 + 2.73i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-5.70 - 9.88i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 35.9T + 1.36e3T^{2} \)
41 \( 1 + (-28.5 + 16.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (11.8 - 20.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (37.1 + 21.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 67.4iT - 2.80e3T^{2} \)
59 \( 1 + (55.9 - 32.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.4 + 47.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-53.6 - 92.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 2.36iT - 5.04e3T^{2} \)
73 \( 1 - 120.T + 5.32e3T^{2} \)
79 \( 1 + (9.18 - 15.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (98.3 + 56.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 24.4iT - 7.92e3T^{2} \)
97 \( 1 + (81.7 - 141. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.37654805964134658817895733852, −14.53616872939223802724756266685, −12.82363924413735364859848478790, −11.83569206396601822242672843480, −10.76365071464509453236305754962, −9.964109674879911491438419400269, −7.982516521247732635864199147383, −6.65190123971351601958535413891, −5.33106204372602687977407002695, −3.84591046438867546990088419300, 0.01448914750047648056155704665, 4.05010582761070464241655478154, 4.80127129433877679976660458064, 7.04383893757139354982417694521, 8.037024655465408787888352198782, 9.414972664163439835351618200925, 11.27099161088238814258326141066, 12.08812182175251669583177435248, 12.56194073043047785423717820282, 13.92880330524225505695823123524

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