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Properties

Label 2-63-7.4-c5-0-11
Degree $2$
Conductor $63$
Sign $-0.0165 + 0.999i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (14 − 24.2i)4-s + (5.5 + 9.52i)5-s + (129.5 − 6.06i)7-s − 120·8-s + (11 − 19.0i)10-s + (134.5 − 232. i)11-s − 308·13-s + (−140 − 218. i)14-s + (−328 − 568. i)16-s + (948 − 1.64e3i)17-s + (82 + 142. i)19-s + 308·20-s − 538·22-s + (−1.63e3 − 2.82e3i)23-s + ⋯
L(s)  = 1  + (−0.176 − 0.306i)2-s + (0.437 − 0.757i)4-s + (0.0983 + 0.170i)5-s + (0.998 − 0.0467i)7-s − 0.662·8-s + (0.0347 − 0.0602i)10-s + (0.335 − 0.580i)11-s − 0.505·13-s + (−0.190 − 0.297i)14-s + (−0.320 − 0.554i)16-s + (0.795 − 1.37i)17-s + (0.0521 + 0.0902i)19-s + 0.172·20-s − 0.236·22-s + (−0.643 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.0165 + 0.999i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.0165 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.25803 - 1.27908i\)
\(L(\frac12)\) \(\approx\) \(1.25803 - 1.27908i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-129.5 + 6.06i)T \)
good2 \( 1 + (1 + 1.73i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-5.5 - 9.52i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-134.5 + 232. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 308T + 3.71e5T^{2} \)
17 \( 1 + (-948 + 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-82 - 142. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.63e3 + 2.82e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.41e3T + 2.05e7T^{2} \)
31 \( 1 + (1.42e3 - 2.46e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-5.66e3 - 9.81e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.68e4T + 1.15e8T^{2} \)
43 \( 1 + 7.89e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.05e4 - 1.82e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.48e4 - 2.57e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (4.08e3 - 7.06e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (7.58e3 + 1.31e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.60e4 + 2.77e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.82e4T + 1.80e9T^{2} \)
73 \( 1 + (1.74e4 - 3.01e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (6.76e3 + 1.17e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.81e4T + 3.93e9T^{2} \)
89 \( 1 + (5.74e4 + 9.95e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.54e5T + 8.58e9T^{2} \)
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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

−14.18486076054878851549714353808, −12.22124514527243934443952069284, −11.33120706085455040414708176564, −10.37413804787813767011511561290, −9.198792087793825889282597150717, −7.69358835229540893068209075687, −6.22089359202062142048700507918, −4.86049214099348166130608297492, −2.60172269683925419811545073202, −0.940870785067477514325087956900, 1.89117142083973604670015071769, 3.89150614055319611921541105177, 5.64254554922747047630528466922, 7.29647728617382301645682747042, 8.117822211397690101190197204966, 9.434732126130794881203252510319, 11.01689256090200368971508731506, 12.03362765745625803400697044795, 12.95214906774863978430227718158, 14.54348893151507141293156939540

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