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Properties

Label 2-63-21.5-c5-0-12
Degree $2$
Conductor $63$
Sign $-0.955 + 0.294i$
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  + (−2.44 − 1.41i)2-s + (−12.0 − 20.7i)4-s + (25.7 − 44.5i)5-s + (122.5 − 42.4i)7-s + 158. i·8-s + (−126 + 72.7i)10-s + (−500. + 289. i)11-s − 1.15e3i·13-s + (−360. − 69.2i)14-s + (−159. + 277. i)16-s + (188. + 326. i)17-s + (−1.50e3 − 866. i)19-s − 1.23e3·20-s + 1.63e3·22-s + (−3.99e3 − 2.30e3i)23-s + ⋯
L(s)  = 1  + (−0.433 − 0.249i)2-s + (−0.375 − 0.649i)4-s + (0.460 − 0.796i)5-s + (0.944 − 0.327i)7-s + 0.874i·8-s + (−0.398 + 0.230i)10-s + (−1.24 + 0.720i)11-s − 1.89i·13-s + (−0.490 − 0.0944i)14-s + (−0.156 + 0.270i)16-s + (0.158 + 0.274i)17-s + (−0.954 − 0.550i)19-s − 0.690·20-s + 0.720·22-s + (−1.57 − 0.908i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.955 + 0.294i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.133448 - 0.887166i\)
\(L(\frac12)\) \(\approx\) \(0.133448 - 0.887166i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-122.5 + 42.4i)T \)
good2 \( 1 + (2.44 + 1.41i)T + (16 + 27.7i)T^{2} \)
5 \( 1 + (-25.7 + 44.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (500. - 289. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.15e3iT - 3.71e5T^{2} \)
17 \( 1 + (-188. - 326. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.50e3 + 866. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (3.99e3 + 2.30e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.60e3iT - 2.05e7T^{2} \)
31 \( 1 + (1.96e3 - 1.13e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (101.5 - 175. i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 85.7T + 1.15e8T^{2} \)
43 \( 1 - 4.69e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.19e4 + 2.07e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-6.42e3 + 3.70e3i)T + (2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.48e3 - 4.30e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.50e4 + 8.71e3i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.50e4 + 2.61e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.96e3iT - 1.80e9T^{2} \)
73 \( 1 + (-4.61e4 + 2.66e4i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.96e4 + 3.41e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 6.35e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.26e4 + 9.12e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 2.06e3iT - 8.58e9T^{2} \)
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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

−13.35574176822331802210330671068, −12.50820855284024271136352926090, −10.63525152233305634153341681090, −10.28155351142116561439315337333, −8.721931433063530567172389068174, −7.86004872286152593505921259162, −5.57692483124641357850947097585, −4.78417563677086150185547734078, −2.02602167781240745596273655563, −0.47162285963257687313774658486, 2.28634691255403143932480141922, 4.20215300055554008749326490045, 6.03435809251748062215670728760, 7.52305291462404807913746577866, 8.483265875717866409645976109435, 9.738571622518777764430809801075, 11.03697327447606394651687935139, 12.13518244125195945335600710903, 13.65043204243891716079263146888, 14.26018136941968912709721274727

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