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Properties

Label 2-63-63.5-c3-0-7
Degree $2$
Conductor $63$
Sign $0.402 - 0.915i$
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  + 0.257i·2-s + (1.70 + 4.90i)3-s + 7.93·4-s + (3.19 + 5.53i)5-s + (−1.26 + 0.439i)6-s + (−15.2 − 10.5i)7-s + 4.10i·8-s + (−21.1 + 16.7i)9-s + (−1.42 + 0.822i)10-s + (52.8 + 30.5i)11-s + (13.5 + 38.9i)12-s + (−8.78 − 5.07i)13-s + (2.72 − 3.91i)14-s + (−21.7 + 25.1i)15-s + 62.4·16-s + (−22.5 − 38.9i)17-s + ⋯
L(s)  = 1  + 0.0910i·2-s + (0.328 + 0.944i)3-s + 0.991·4-s + (0.285 + 0.494i)5-s + (−0.0860 + 0.0299i)6-s + (−0.821 − 0.570i)7-s + 0.181i·8-s + (−0.784 + 0.620i)9-s + (−0.0450 + 0.0260i)10-s + (1.44 + 0.837i)11-s + (0.325 + 0.936i)12-s + (−0.187 − 0.108i)13-s + (0.0519 − 0.0748i)14-s + (−0.373 + 0.432i)15-s + 0.975·16-s + (−0.321 − 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.402 - 0.915i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.402 - 0.915i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.55099 + 1.01289i\)
\(L(\frac12)\) \(\approx\) \(1.55099 + 1.01289i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 4.90i)T \)
7 \( 1 + (15.2 + 10.5i)T \)
good2 \( 1 - 0.257iT - 8T^{2} \)
5 \( 1 + (-3.19 - 5.53i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-52.8 - 30.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (8.78 + 5.07i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (22.5 + 38.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.6 + 40.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-23.9 + 13.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-48.9 + 28.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 106. iT - 2.97e4T^{2} \)
37 \( 1 + (-95.6 + 165. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-15.0 + 26.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (185. + 320. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 496.T + 1.03e5T^{2} \)
53 \( 1 + (601. - 347. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 635.T + 2.05e5T^{2} \)
61 \( 1 - 747. iT - 2.26e5T^{2} \)
67 \( 1 - 164.T + 3.00e5T^{2} \)
71 \( 1 - 278. iT - 3.57e5T^{2} \)
73 \( 1 + (313. - 181. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 557.T + 4.93e5T^{2} \)
83 \( 1 + (514. + 891. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (730. - 1.26e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (878. - 507. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.84229987744946196339761616134, −13.90793567027805295243826723783, −12.29039493703232725548685176220, −11.02102436572709767746273875716, −10.14536994729970764938010347178, −9.116948565383344750065061555161, −7.22703756916681949086105192558, −6.26258944246564064663632092658, −4.17631850454561644089156072513, −2.61407081398312776366130064844, 1.54237339489147252971021674566, 3.22345684646662084287761381498, 6.07619496922499911698686633768, 6.68653709803041805929196374803, 8.363852098558518822471775187426, 9.396455860282445516732472927702, 11.15570079513719353972406002559, 12.21662143416932775142278317918, 12.89117402480387076018471459879, 14.22107094237361842061315352976

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