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Properties

Label 2-63-63.38-c3-0-4
Degree $2$
Conductor $63$
Sign $-0.477 - 0.878i$
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  + 2.41i·2-s + (−3.31 − 4.00i)3-s + 2.16·4-s + (−5.35 + 9.28i)5-s + (9.67 − 8.00i)6-s + (3.86 + 18.1i)7-s + 24.5i·8-s + (−5.05 + 26.5i)9-s + (−22.4 − 12.9i)10-s + (−22.6 + 13.0i)11-s + (−7.15 − 8.64i)12-s + (29.8 − 17.2i)13-s + (−43.7 + 9.33i)14-s + (54.9 − 9.29i)15-s − 42.0·16-s + (−2.29 + 3.98i)17-s + ⋯
L(s)  = 1  + 0.854i·2-s + (−0.637 − 0.770i)3-s + 0.270·4-s + (−0.479 + 0.830i)5-s + (0.658 − 0.544i)6-s + (0.208 + 0.978i)7-s + 1.08i·8-s + (−0.187 + 0.982i)9-s + (−0.709 − 0.409i)10-s + (−0.620 + 0.358i)11-s + (−0.172 − 0.208i)12-s + (0.637 − 0.368i)13-s + (−0.835 + 0.178i)14-s + (0.945 − 0.159i)15-s − 0.657·16-s + (−0.0327 + 0.0567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.477 - 0.878i$
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: \((2,\ 63,\ (\ :3/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.567450 + 0.953810i\)
\(L(\frac12)\) \(\approx\) \(0.567450 + 0.953810i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.31 + 4.00i)T \)
7 \( 1 + (-3.86 - 18.1i)T \)
good2 \( 1 - 2.41iT - 8T^{2} \)
5 \( 1 + (5.35 - 9.28i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (22.6 - 13.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-29.8 + 17.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (2.29 - 3.98i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-6.74 + 3.89i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (33.9 + 19.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-210. - 121. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 311. iT - 2.97e4T^{2} \)
37 \( 1 + (28.3 + 49.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (148. + 256. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (60.7 - 105. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 225.T + 1.03e5T^{2} \)
53 \( 1 + (-307. - 177. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 271.T + 2.05e5T^{2} \)
61 \( 1 - 749. iT - 2.26e5T^{2} \)
67 \( 1 - 994.T + 3.00e5T^{2} \)
71 \( 1 + 1.13e3iT - 3.57e5T^{2} \)
73 \( 1 + (-1.07e3 - 620. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 433.T + 4.93e5T^{2} \)
83 \( 1 + (492. - 853. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (75.6 + 131. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (781. + 451. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−15.15695919582856469819052237947, −13.91756521369825222292089558387, −12.47463125554911405195936197981, −11.49338990495853792617822853206, −10.64550184927911598451325091865, −8.381744071175914740044448502428, −7.45262435428018917463098140420, −6.38353789712902397976676888872, −5.35429269017135668678125789639, −2.46580445501006281520513080518, 0.847961582973718987215475535114, 3.59059816956902647537424703465, 4.79093987516211689430157602071, 6.62649403121557837913019592214, 8.385339309917734958738779624865, 9.949421041223564867846391238217, 10.77846970000065845609669488008, 11.66210220669844138847498938268, 12.59101745278538340471726594812, 13.90068620086077476535780569167

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