↑

Properties

Label 2-26-13.8-c2-0-0
Degree 22
Conductor 2626
Sign 0.881−0.471i0.881 - 0.471i
Analytic cond. 0.7084480.708448
Root an. cond. 0.8416930.841693
Motivic weight 22
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 + i)2-s + 2i·4-s + (−3 − 3i)5-s + (2 − 2i)7-s + (−2 + 2i)8-s − 9·9-s − 6i·10-s + (6 − 6i)11-s + 13i·13-s + 4·14-s − 4·16-s + 6i·17-s + (−9 − 9i)18-s + (26 + 26i)19-s + (6 − 6i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.600 − 0.600i)5-s + (0.285 − 0.285i)7-s + (−0.250 + 0.250i)8-s − 9-s − 0.600i·10-s + (0.545 − 0.545i)11-s + i·13-s + 0.285·14-s − 0.250·16-s + 0.352i·17-s + (−0.5 − 0.5i)18-s + (1.36 + 1.36i)19-s + (0.300 − 0.300i)20-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((0.881−0.471i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+1) L(s)=((0.881−0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: 0.881−0.471i0.881 - 0.471i
Analytic conductor: 0.7084480.708448
Root analytic conductor: 0.8416930.841693
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ26(21,⋅)\chi_{26} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :1), 0.881−0.471i)(2,\ 26,\ (\ :1),\ 0.881 - 0.471i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.05219+0.263852i1.05219 + 0.263852i
L(12)L(\frac12) ≈\approx 1.05219+0.263852i1.05219 + 0.263852i
L(2)L(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−i)T 1 + (-1 - i)T
13 1−13iT 1 - 13iT
good3 1+9T2 1 + 9T^{2}
5 1+(3+3i)T+25iT2 1 + (3 + 3i)T + 25iT^{2}
7 1+(−2+2i)T−49iT2 1 + (-2 + 2i)T - 49iT^{2}
11 1+(−6+6i)T−121iT2 1 + (-6 + 6i)T - 121iT^{2}
17 1−6iT−289T2 1 - 6iT - 289T^{2}
19 1+(−26−26i)T+361iT2 1 + (-26 - 26i)T + 361iT^{2}
23 1+24iT−529T2 1 + 24iT - 529T^{2}
29 1+48T+841T2 1 + 48T + 841T^{2}
31 1+(14+14i)T+961iT2 1 + (14 + 14i)T + 961iT^{2}
37 1+(−37+37i)T−1.36e3iT2 1 + (-37 + 37i)T - 1.36e3iT^{2}
41 1+(9+9i)T+1.68e3iT2 1 + (9 + 9i)T + 1.68e3iT^{2}
43 1−36iT−1.84e3T2 1 - 36iT - 1.84e3T^{2}
47 1+(−42+42i)T−2.20e3iT2 1 + (-42 + 42i)T - 2.20e3iT^{2}
53 1−30T+2.80e3T2 1 - 30T + 2.80e3T^{2}
59 1+(54−54i)T−3.48e3iT2 1 + (54 - 54i)T - 3.48e3iT^{2}
61 1+18T+3.72e3T2 1 + 18T + 3.72e3T^{2}
67 1+(22+22i)T+4.48e3iT2 1 + (22 + 22i)T + 4.48e3iT^{2}
71 1+(−6−6i)T+5.04e3iT2 1 + (-6 - 6i)T + 5.04e3iT^{2}
73 1+(−17+17i)T−5.32e3iT2 1 + (-17 + 17i)T - 5.32e3iT^{2}
79 1+108T+6.24e3T2 1 + 108T + 6.24e3T^{2}
83 1+(−78−78i)T+6.88e3iT2 1 + (-78 - 78i)T + 6.88e3iT^{2}
89 1+(9−9i)T−7.92e3iT2 1 + (9 - 9i)T - 7.92e3iT^{2}
97 1+(47+47i)T+9.40e3iT2 1 + (47 + 47i)T + 9.40e3iT^{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

−16.75075000834519460076311928876, −16.44712681731140219534814499481, −14.70294232960147805991572306730, −13.90300393988905715008427178251, −12.27189113398734597220008821566, −11.30287356242241452065891629106, −9.001743277187219891262692603270, −7.74904870654413418110336879363, −5.84602179849635622379922007877, −4.00527522059654166627719770843, 3.18651302706180249264320587061, 5.38569459593678042418592606655, 7.41588762584680208753980303552, 9.331167023136773648796773316813, 11.12671721156407251343402286710, 11.80008865402941539056754557416, 13.41921440535433854909104938686, 14.72775324191117174237516499556, 15.50405230739533236134390730981, 17.35449125857574115996580014573

Graph of the ZZ-function along the critical line