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Properties

Label 2-63-63.23-c2-0-0
Degree $2$
Conductor $63$
Sign $-0.915 + 0.401i$
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  + 3.22i·2-s + (−2.32 − 1.89i)3-s − 6.39·4-s + (−4.79 + 2.76i)5-s + (6.10 − 7.50i)6-s + (−6.99 − 0.206i)7-s − 7.70i·8-s + (1.82 + 8.81i)9-s + (−8.91 − 15.4i)10-s + (15.3 + 8.84i)11-s + (14.8 + 12.1i)12-s + (2.03 − 3.52i)13-s + (0.665 − 22.5i)14-s + (16.3 + 2.63i)15-s − 0.715·16-s + (−14.3 + 8.27i)17-s + ⋯
L(s)  = 1  + 1.61i·2-s + (−0.775 − 0.631i)3-s − 1.59·4-s + (−0.958 + 0.553i)5-s + (1.01 − 1.25i)6-s + (−0.999 − 0.0294i)7-s − 0.963i·8-s + (0.203 + 0.979i)9-s + (−0.891 − 1.54i)10-s + (1.39 + 0.804i)11-s + (1.23 + 1.00i)12-s + (0.156 − 0.271i)13-s + (0.0475 − 1.61i)14-s + (1.09 + 0.175i)15-s − 0.0447·16-s + (−0.843 + 0.486i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.915 + 0.401i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.915 + 0.401i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0982891 - 0.469184i\)
\(L(\frac12)\) \(\approx\) \(0.0982891 - 0.469184i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.32 + 1.89i)T \)
7 \( 1 + (6.99 + 0.206i)T \)
good2 \( 1 - 3.22iT - 4T^{2} \)
5 \( 1 + (4.79 - 2.76i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-15.3 - 8.84i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.03 + 3.52i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (14.3 - 8.27i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3.92 - 6.79i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.71 + 5.03i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (39.9 - 23.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 29.6T + 961T^{2} \)
37 \( 1 + (-15.5 + 27.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-27.8 - 16.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.35 - 5.80i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 16.4iT - 2.20e3T^{2} \)
53 \( 1 + (32.5 - 18.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 95.0iT - 3.48e3T^{2} \)
61 \( 1 + 73.7T + 3.72e3T^{2} \)
67 \( 1 + 12.1T + 4.48e3T^{2} \)
71 \( 1 - 20.0iT - 5.04e3T^{2} \)
73 \( 1 + (11.4 + 19.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 138.T + 6.24e3T^{2} \)
83 \( 1 + (-13.6 + 7.90i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (46.9 + 27.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-86.1 - 149. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.42545571160667637399947059231, −14.64084946730940203716364168577, −13.27260012910896755059679621080, −12.21544173031453596940561468827, −10.94514640011731605373566170680, −9.177291351558186116172600023674, −7.65306353522304996048005953645, −6.84198526327876333673349553119, −6.08671492236474833199556270739, −4.23145641032519137339379246079, 0.49399152011033700893973996607, 3.52656214496322384633372660971, 4.41644864369809905289471243342, 6.46593421671320551856048375501, 8.944481569715082385164913646858, 9.617743818522722925979577161089, 11.09873643314263035563508126734, 11.62539484095168217076057378175, 12.47301022923135225109538062794, 13.59425372436257652903050226272

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