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Properties

Label 2-63-63.41-c3-0-12
Degree $2$
Conductor $63$
Sign $0.606 + 0.795i$
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  + (−0.0847 + 0.0489i)2-s + (−5.17 + 0.490i)3-s + (−3.99 + 6.91i)4-s + (9.06 − 15.6i)5-s + (0.414 − 0.294i)6-s + (12.7 − 13.4i)7-s − 1.56i·8-s + (26.5 − 5.07i)9-s + 1.77i·10-s + (32.0 − 18.5i)11-s + (17.2 − 37.7i)12-s + (−16.3 − 9.44i)13-s + (−0.423 + 1.76i)14-s + (−39.1 + 85.6i)15-s + (−31.8 − 55.2i)16-s − 62.5·17-s + ⋯
L(s)  = 1  + (−0.0299 + 0.0173i)2-s + (−0.995 + 0.0944i)3-s + (−0.499 + 0.864i)4-s + (0.810 − 1.40i)5-s + (0.0282 − 0.0200i)6-s + (0.688 − 0.725i)7-s − 0.0691i·8-s + (0.982 − 0.188i)9-s + 0.0561i·10-s + (0.878 − 0.507i)11-s + (0.415 − 0.908i)12-s + (−0.349 − 0.201i)13-s + (−0.00808 + 0.0336i)14-s + (−0.674 + 1.47i)15-s + (−0.498 − 0.862i)16-s − 0.891·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.606 + 0.795i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.606 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.971533 - 0.480956i\)
\(L(\frac12)\) \(\approx\) \(0.971533 - 0.480956i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.17 - 0.490i)T \)
7 \( 1 + (-12.7 + 13.4i)T \)
good2 \( 1 + (0.0847 - 0.0489i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-9.06 + 15.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-32.0 + 18.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (16.3 + 9.44i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 62.5T + 4.91e3T^{2} \)
19 \( 1 + 70.2iT - 6.85e3T^{2} \)
23 \( 1 + (-140. - 81.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-82.5 + 47.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-110. - 63.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 378.T + 5.06e4T^{2} \)
41 \( 1 + (-99.1 + 171. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-160. - 278. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-79.3 - 137. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 191. iT - 1.48e5T^{2} \)
59 \( 1 + (106. - 185. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (190. - 110. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-68.2 + 118. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 458. iT - 3.57e5T^{2} \)
73 \( 1 - 967. iT - 3.89e5T^{2} \)
79 \( 1 + (-298. - 516. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-180. - 313. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 35.4T + 7.04e5T^{2} \)
97 \( 1 + (1.15e3 - 664. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.85448589421654221760514879611, −13.13012383771828051200600650491, −12.16217578031367397755189532477, −11.08238935376811965653907770348, −9.500082782817463994918746810033, −8.588292615387994117241072876879, −6.96559325325041144428388622619, −5.19175296937739259225732524127, −4.34255088887471091781662882816, −0.962658406447336003043529641479, 1.87832513603631104453891119308, 4.76797096637248047879095115633, 6.04700371295315449287247004729, 6.88010687974091640042812453652, 9.137743715627905258186427074496, 10.29610110051044832740045139926, 11.00346890108549914437757426968, 12.20370316910533525145840920070, 13.76562950215526166385856017888, 14.65824456885929173969741973893

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