↑

Properties

Label 2-117-117.11-c1-0-1
Degree 22
Conductor 117117
Sign 0.378−0.925i0.378 - 0.925i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.75 + 1.75i)2-s + (−1.38 − 1.03i)3-s − 4.17i·4-s + (3.52 + 0.943i)5-s + (4.25 − 0.617i)6-s + (−1.05 − 0.281i)7-s + (3.81 + 3.81i)8-s + (0.852 + 2.87i)9-s + (−7.84 + 4.52i)10-s + (1.46 + 1.46i)11-s + (−4.32 + 5.78i)12-s + (3.60 + 0.0594i)13-s + (2.34 − 1.35i)14-s + (−3.90 − 4.95i)15-s − 5.05·16-s + (−2.40 + 4.17i)17-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)2-s + (−0.801 − 0.598i)3-s − 2.08i·4-s + (1.57 + 0.421i)5-s + (1.73 − 0.252i)6-s + (−0.397 − 0.106i)7-s + (1.34 + 1.34i)8-s + (0.284 + 0.958i)9-s + (−2.47 + 1.43i)10-s + (0.442 + 0.442i)11-s + (−1.24 + 1.67i)12-s + (0.999 + 0.0164i)13-s + (0.626 − 0.361i)14-s + (−1.00 − 1.27i)15-s − 1.26·16-s + (−0.584 + 1.01i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.378−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.378−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.378−0.925i0.378 - 0.925i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(11,⋅)\chi_{117} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.378−0.925i)(2,\ 117,\ (\ :1/2),\ 0.378 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.481460+0.323407i0.481460 + 0.323407i
L(12)L(\frac12) ≈\approx 0.481460+0.323407i0.481460 + 0.323407i
L(32)L(\frac{3}{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)
bad3 1+(1.38+1.03i)T 1 + (1.38 + 1.03i)T
13 1+(−3.60−0.0594i)T 1 + (-3.60 - 0.0594i)T
good2 1+(1.75−1.75i)T−2iT2 1 + (1.75 - 1.75i)T - 2iT^{2}
5 1+(−3.52−0.943i)T+(4.33+2.5i)T2 1 + (-3.52 - 0.943i)T + (4.33 + 2.5i)T^{2}
7 1+(1.05+0.281i)T+(6.06+3.5i)T2 1 + (1.05 + 0.281i)T + (6.06 + 3.5i)T^{2}
11 1+(−1.46−1.46i)T+11iT2 1 + (-1.46 - 1.46i)T + 11iT^{2}
17 1+(2.40−4.17i)T+(−8.5−14.7i)T2 1 + (2.40 - 4.17i)T + (-8.5 - 14.7i)T^{2}
19 1+(−5.01+1.34i)T+(16.4−9.5i)T2 1 + (-5.01 + 1.34i)T + (16.4 - 9.5i)T^{2}
23 1+(−0.496+0.860i)T+(−11.5−19.9i)T2 1 + (-0.496 + 0.860i)T + (-11.5 - 19.9i)T^{2}
29 1+3.59iT−29T2 1 + 3.59iT - 29T^{2}
31 1+(−0.832+3.10i)T+(−26.8−15.5i)T2 1 + (-0.832 + 3.10i)T + (-26.8 - 15.5i)T^{2}
37 1+(3.11+0.834i)T+(32.0+18.5i)T2 1 + (3.11 + 0.834i)T + (32.0 + 18.5i)T^{2}
41 1+(0.0289+0.108i)T+(−35.5+20.5i)T2 1 + (0.0289 + 0.108i)T + (-35.5 + 20.5i)T^{2}
43 1+(7.98−4.61i)T+(21.5−37.2i)T2 1 + (7.98 - 4.61i)T + (21.5 - 37.2i)T^{2}
47 1+(7.57−2.02i)T+(40.7−23.5i)T2 1 + (7.57 - 2.02i)T + (40.7 - 23.5i)T^{2}
53 1+9.48iT−53T2 1 + 9.48iT - 53T^{2}
59 1+(−0.399−0.399i)T+59iT2 1 + (-0.399 - 0.399i)T + 59iT^{2}
61 1+(−4.06−7.03i)T+(−30.5+52.8i)T2 1 + (-4.06 - 7.03i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.34+1.43i)T+(58.0−33.5i)T2 1 + (-5.34 + 1.43i)T + (58.0 - 33.5i)T^{2}
71 1+(0.773+2.88i)T+(−61.4+35.5i)T2 1 + (0.773 + 2.88i)T + (-61.4 + 35.5i)T^{2}
73 1+(−4.89+4.89i)T−73iT2 1 + (-4.89 + 4.89i)T - 73iT^{2}
79 1+(5.66−9.80i)T+(−39.5−68.4i)T2 1 + (5.66 - 9.80i)T + (-39.5 - 68.4i)T^{2}
83 1+(−0.638−2.38i)T+(−71.8+41.5i)T2 1 + (-0.638 - 2.38i)T + (-71.8 + 41.5i)T^{2}
89 1+(−0.0437+0.163i)T+(−77.0−44.5i)T2 1 + (-0.0437 + 0.163i)T + (-77.0 - 44.5i)T^{2}
97 1+(−1.80+6.75i)T+(−84.0−48.5i)T2 1 + (-1.80 + 6.75i)T + (-84.0 - 48.5i)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

−13.78906149659724773919716147713, −13.04866536851465901168757940187, −11.25070930917446704697419848966, −10.17834483972844572378804888920, −9.534241322988936873619259183412, −8.233595029901006037449580233618, −6.76752142275052530490703079959, −6.38457167427000061768185465682, −5.42050508945218294715953606913, −1.59609120152007447449190606302, 1.31935030372453493567548881692, 3.28227280892103264312737790922, 5.31176400200569388108379584681, 6.61140149039016115418631610230, 8.745962698586330733749900804166, 9.421892172865802395803539726328, 10.07254066440050213481032819139, 11.04504552501214485179387906788, 11.90514079846360584518877703436, 12.99477532183159590566257946351

Graph of the ZZ-function along the critical line