↑

Properties

Label 2-368-16.13-c1-0-15
Degree 22
Conductor 368368
Sign 0.835−0.549i0.835 - 0.549i
Analytic cond. 2.938492.93849
Root an. cond. 1.714201.71420
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  + (−1.27 + 0.602i)2-s + (0.203 + 0.203i)3-s + (1.27 − 1.54i)4-s + (1.88 − 1.88i)5-s + (−0.383 − 0.137i)6-s + 4.71i·7-s + (−0.702 + 2.73i)8-s − 2.91i·9-s + (−1.27 + 3.54i)10-s + (−0.944 + 0.944i)11-s + (0.573 − 0.0543i)12-s + (1.72 + 1.72i)13-s + (−2.83 − 6.03i)14-s + 0.766·15-s + (−0.751 − 3.92i)16-s + 7.35·17-s + ⋯
L(s)  = 1  + (−0.904 + 0.425i)2-s + (0.117 + 0.117i)3-s + (0.637 − 0.770i)4-s + (0.841 − 0.841i)5-s + (−0.156 − 0.0562i)6-s + 1.78i·7-s + (−0.248 + 0.968i)8-s − 0.972i·9-s + (−0.402 + 1.11i)10-s + (−0.284 + 0.284i)11-s + (0.165 − 0.0156i)12-s + (0.478 + 0.478i)13-s + (−0.758 − 1.61i)14-s + 0.197·15-s + (−0.187 − 0.982i)16-s + 1.78·17-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((0.835−0.549i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((0.835−0.549i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: 0.835−0.549i0.835 - 0.549i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(93,⋅)\chi_{368} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), 0.835−0.549i)(2,\ 368,\ (\ :1/2),\ 0.835 - 0.549i)

Particular Values

L(1)L(1) ≈\approx 1.06407+0.318567i1.06407 + 0.318567i
L(12)L(\frac12) ≈\approx 1.06407+0.318567i1.06407 + 0.318567i
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)
bad2 1+(1.27−0.602i)T 1 + (1.27 - 0.602i)T
23 1+iT 1 + iT
good3 1+(−0.203−0.203i)T+3iT2 1 + (-0.203 - 0.203i)T + 3iT^{2}
5 1+(−1.88+1.88i)T−5iT2 1 + (-1.88 + 1.88i)T - 5iT^{2}
7 1−4.71iT−7T2 1 - 4.71iT - 7T^{2}
11 1+(0.944−0.944i)T−11iT2 1 + (0.944 - 0.944i)T - 11iT^{2}
13 1+(−1.72−1.72i)T+13iT2 1 + (-1.72 - 1.72i)T + 13iT^{2}
17 1−7.35T+17T2 1 - 7.35T + 17T^{2}
19 1+(1.62+1.62i)T+19iT2 1 + (1.62 + 1.62i)T + 19iT^{2}
29 1+(1.16+1.16i)T+29iT2 1 + (1.16 + 1.16i)T + 29iT^{2}
31 1−9.98T+31T2 1 - 9.98T + 31T^{2}
37 1+(3.26−3.26i)T−37iT2 1 + (3.26 - 3.26i)T - 37iT^{2}
41 1−5.41iT−41T2 1 - 5.41iT - 41T^{2}
43 1+(5.65−5.65i)T−43iT2 1 + (5.65 - 5.65i)T - 43iT^{2}
47 1−1.93T+47T2 1 - 1.93T + 47T^{2}
53 1+(−7.79+7.79i)T−53iT2 1 + (-7.79 + 7.79i)T - 53iT^{2}
59 1+(−6.83+6.83i)T−59iT2 1 + (-6.83 + 6.83i)T - 59iT^{2}
61 1+(−1.70−1.70i)T+61iT2 1 + (-1.70 - 1.70i)T + 61iT^{2}
67 1+(9.77+9.77i)T+67iT2 1 + (9.77 + 9.77i)T + 67iT^{2}
71 1−0.932iT−71T2 1 - 0.932iT - 71T^{2}
73 1−3.71iT−73T2 1 - 3.71iT - 73T^{2}
79 1+10.7T+79T2 1 + 10.7T + 79T^{2}
83 1+(4.25+4.25i)T+83iT2 1 + (4.25 + 4.25i)T + 83iT^{2}
89 1+0.581iT−89T2 1 + 0.581iT - 89T^{2}
97 1+9.37T+97T2 1 + 9.37T + 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

−11.65988317452632353201690123643, −9.990792647802070196752771965341, −9.583818140395529701128172353729, −8.733601900967680798051110093630, −8.215672520435895241346336199859, −6.55130571583917741727416437485, −5.83230248902318969004168828300, −5.03670642108971078905327971635, −2.81082021004663622344326400844, −1.44394112667617593768511346955, 1.24750137140108481252106384860, 2.76229636406048027532304626742, 3.86457519927869012694671018940, 5.70205442743092412033065746288, 6.94959718508441058541689989599, 7.61031468710999675672671492461, 8.415476074694278076018699298310, 10.01985772414975325018763903018, 10.36692158061112043235457888494, 10.74321091354257789681941862757

Graph of the ZZ-function along the critical line