↑

Properties

Label 2-624-13.3-c3-0-17
Degree 22
Conductor 624624
Sign −0.0942−0.995i-0.0942 - 0.995i
Analytic cond. 36.817136.8171
Root an. cond. 6.067716.06771
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)3-s + 15.3·5-s + (1.63 − 2.82i)7-s + (−4.5 + 7.79i)9-s + (−24.6 − 42.7i)11-s + (−8.10 + 46.1i)13-s + (23.0 + 39.8i)15-s + (−53.6 + 92.8i)17-s + (−59.9 + 103. i)19-s + 9.80·21-s + (15.2 + 26.3i)23-s + 110.·25-s − 27·27-s + (86.5 + 149. i)29-s + 189.·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 1.37·5-s + (0.0882 − 0.152i)7-s + (−0.166 + 0.288i)9-s + (−0.676 − 1.17i)11-s + (−0.172 + 0.984i)13-s + (0.396 + 0.686i)15-s + (−0.764 + 1.32i)17-s + (−0.724 + 1.25i)19-s + 0.101·21-s + (0.137 + 0.239i)23-s + 0.886·25-s − 0.192·27-s + (0.554 + 0.960i)29-s + 1.09·31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.0942−0.995i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((−0.0942−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.0942−0.995i-0.0942 - 0.995i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(289,⋅)\chi_{624} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), −0.0942−0.995i)(2,\ 624,\ (\ :3/2),\ -0.0942 - 0.995i)

Particular Values

L(2)L(2) ≈\approx 2.3275062682.327506268
L(12)L(\frac12) ≈\approx 2.3275062682.327506268
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(−1.5−2.59i)T 1 + (-1.5 - 2.59i)T
13 1+(8.10−46.1i)T 1 + (8.10 - 46.1i)T
good5 1−15.3T+125T2 1 - 15.3T + 125T^{2}
7 1+(−1.63+2.82i)T+(−171.5−297.i)T2 1 + (-1.63 + 2.82i)T + (-171.5 - 297. i)T^{2}
11 1+(24.6+42.7i)T+(−665.5+1.15e3i)T2 1 + (24.6 + 42.7i)T + (-665.5 + 1.15e3i)T^{2}
17 1+(53.6−92.8i)T+(−2.45e3−4.25e3i)T2 1 + (53.6 - 92.8i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(59.9−103.i)T+(−3.42e3−5.94e3i)T2 1 + (59.9 - 103. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−15.2−26.3i)T+(−6.08e3+1.05e4i)T2 1 + (-15.2 - 26.3i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−86.5−149.i)T+(−1.21e4+2.11e4i)T2 1 + (-86.5 - 149. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1−189.T+2.97e4T2 1 - 189.T + 2.97e4T^{2}
37 1+(−155.−269.i)T+(−2.53e4+4.38e4i)T2 1 + (-155. - 269. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+(65.2+113.i)T+(−3.44e4+5.96e4i)T2 1 + (65.2 + 113. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(21.4−37.0i)T+(−3.97e4−6.88e4i)T2 1 + (21.4 - 37.0i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+336.T+1.03e5T2 1 + 336.T + 1.03e5T^{2}
53 1−673.T+1.48e5T2 1 - 673.T + 1.48e5T^{2}
59 1+(83.2−144.i)T+(−1.02e5−1.77e5i)T2 1 + (83.2 - 144. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−268.+464.i)T+(−1.13e5−1.96e5i)T2 1 + (-268. + 464. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(233.+405.i)T+(−1.50e5+2.60e5i)T2 1 + (233. + 405. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+(360.−624.i)T+(−1.78e5−3.09e5i)T2 1 + (360. - 624. i)T + (-1.78e5 - 3.09e5i)T^{2}
73 1−408.T+3.89e5T2 1 - 408.T + 3.89e5T^{2}
79 1−1.09e3T+4.93e5T2 1 - 1.09e3T + 4.93e5T^{2}
83 1−1.15e3T+5.71e5T2 1 - 1.15e3T + 5.71e5T^{2}
89 1+(683.+1.18e3i)T+(−3.52e5+6.10e5i)T2 1 + (683. + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+(559.−969.i)T+(−4.56e5−7.90e5i)T2 1 + (559. - 969. i)T + (-4.56e5 - 7.90e5i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.41029735119839418695113896152, −9.662009468330108467508709885226, −8.690690594564922188174885667177, −8.154202135369727532309701295156, −6.54368026612706708342459600088, −5.99195759646267567917338731386, −4.94892576861295723098000931081, −3.82475573400228393341752659675, −2.55331493437110696052465700136, −1.51795067716779796728333925390, 0.61094149988249476526194285577, 2.35293768778400950626149670103, 2.52378132460520245529355317828, 4.60932640433738921532210191719, 5.38349497249315117780707349978, 6.47971892412206365843817733264, 7.19925912388504837367840658868, 8.245307177014464091164853418495, 9.230892038629014049169721176401, 9.884089950398394992790986458618

Graph of the ZZ-function along the critical line