↑

Properties

Label 2-325-65.47-c1-0-8
Degree 22
Conductor 325325
Sign 0.966+0.256i0.966 + 0.256i
Analytic cond. 2.595132.59513
Root an. cond. 1.610941.61094
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  − i·2-s + (−1 + i)3-s + 4-s + (1 + i)6-s + 2·7-s − 3i·8-s + i·9-s + (−1 + i)11-s + (−1 + i)12-s + (2 + 3i)13-s − 2i·14-s − 16-s + (1 − i)17-s + 18-s + (5 − 5i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (0.408 + 0.408i)6-s + 0.755·7-s − 1.06i·8-s + 0.333i·9-s + (−0.301 + 0.301i)11-s + (−0.288 + 0.288i)12-s + (0.554 + 0.832i)13-s − 0.534i·14-s − 0.250·16-s + (0.242 − 0.242i)17-s + 0.235·18-s + (1.14 − 1.14i)19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((0.966+0.256i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+1/2) L(s)=((0.966+0.256i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: 0.966+0.256i0.966 + 0.256i
Analytic conductor: 2.595132.59513
Root analytic conductor: 1.610941.61094
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ325(307,⋅)\chi_{325} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :1/2), 0.966+0.256i)(2,\ 325,\ (\ :1/2),\ 0.966 + 0.256i)

Particular Values

L(1)L(1) ≈\approx 1.39869−0.182557i1.39869 - 0.182557i
L(12)L(\frac12) ≈\approx 1.39869−0.182557i1.39869 - 0.182557i
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)
bad5 1 1
13 1+(−2−3i)T 1 + (-2 - 3i)T
good2 1+iT−2T2 1 + iT - 2T^{2}
3 1+(1−i)T−3iT2 1 + (1 - i)T - 3iT^{2}
7 1−2T+7T2 1 - 2T + 7T^{2}
11 1+(1−i)T−11iT2 1 + (1 - i)T - 11iT^{2}
17 1+(−1+i)T−17iT2 1 + (-1 + i)T - 17iT^{2}
19 1+(−5+5i)T−19iT2 1 + (-5 + 5i)T - 19iT^{2}
23 1+(−3−3i)T+23iT2 1 + (-3 - 3i)T + 23iT^{2}
29 1−29T2 1 - 29T^{2}
31 1+(−5−5i)T+31iT2 1 + (-5 - 5i)T + 31iT^{2}
37 1+37T2 1 + 37T^{2}
41 1+(7+7i)T+41iT2 1 + (7 + 7i)T + 41iT^{2}
43 1+(1+i)T+43iT2 1 + (1 + i)T + 43iT^{2}
47 1+6T+47T2 1 + 6T + 47T^{2}
53 1+(5−5i)T−53iT2 1 + (5 - 5i)T - 53iT^{2}
59 1+(−7−7i)T+59iT2 1 + (-7 - 7i)T + 59iT^{2}
61 1+14T+61T2 1 + 14T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+(−1−i)T+71iT2 1 + (-1 - i)T + 71iT^{2}
73 1+10iT−73T2 1 + 10iT - 73T^{2}
79 1+2iT−79T2 1 + 2iT - 79T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+(5+5i)T+89iT2 1 + (5 + 5i)T + 89iT^{2}
97 1+2iT−97T2 1 + 2iT - 97T^{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

−11.49877995026866319420390841576, −10.82669726838418579362701549665, −10.05127028613271994248561309330, −9.050008011406213315356624561879, −7.67109668290368276829886412309, −6.75376157360806958794585756715, −5.35802131020787838166673963893, −4.50679979538928923128421638146, −3.05605240781393840939190521912, −1.56913327172388475004175740625, 1.38698629632226783143308340965, 3.20387542892187851939078231485, 5.10467461181155124744900307917, 5.92458878441165171090142572011, 6.68892470576552616216444542324, 7.87155574364968762775767305140, 8.253305675688649284095761861164, 9.873443212325966159410298061836, 11.01293290014164402462558776367, 11.59263846417408964018128818610

Graph of the ZZ-function along the critical line