Properties

Label 2-26e2-13.3-c1-0-3
Degree 22
Conductor 676676
Sign 0.01280.999i0.0128 - 0.999i
Analytic cond. 5.397885.39788
Root an. cond. 2.323332.32333
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)3-s + 3·5-s + (−2 + 3.46i)7-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)15-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)19-s − 7.99·21-s + (3 + 5.19i)23-s + 4·25-s + 4.00·27-s + (−4.5 − 7.79i)29-s − 2·31-s + (−6 + 10.3i)35-s + (−3.5 − 6.06i)37-s + ⋯
L(s)  = 1  + (0.577 + 0.999i)3-s + 1.34·5-s + (−0.755 + 1.30i)7-s + (−0.166 + 0.288i)9-s + (0.774 + 1.34i)15-s + (−0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s − 1.74·21-s + (0.625 + 1.08i)23-s + 0.800·25-s + 0.769·27-s + (−0.835 − 1.44i)29-s − 0.359·31-s + (−1.01 + 1.75i)35-s + (−0.575 − 0.996i)37-s + ⋯

Functional equation

Λ(s)=(676s/2ΓC(s)L(s)=((0.01280.999i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(676s/2ΓC(s+1/2)L(s)=((0.01280.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 676676    =    221322^{2} \cdot 13^{2}
Sign: 0.01280.999i0.0128 - 0.999i
Analytic conductor: 5.397885.39788
Root analytic conductor: 2.323332.32333
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ676(653,)\chi_{676} (653, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 676, ( :1/2), 0.01280.999i)(2,\ 676,\ (\ :1/2),\ 0.0128 - 0.999i)

Particular Values

L(1)L(1) \approx 1.47685+1.45804i1.47685 + 1.45804i
L(12)L(\frac12) \approx 1.47685+1.45804i1.47685 + 1.45804i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
13 1 1
good3 1+(11.73i)T+(1.5+2.59i)T2 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2}
5 13T+5T2 1 - 3T + 5T^{2}
7 1+(23.46i)T+(3.56.06i)T2 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2}
11 1+(5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
17 1+(1.52.59i)T+(8.514.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(1+1.73i)T+(9.516.4i)T2 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1+(35.19i)T+(11.5+19.9i)T2 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.5+7.79i)T+(14.5+25.1i)T2 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+(3.5+6.06i)T+(18.5+32.0i)T2 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.52.59i)T+(20.5+35.5i)T2 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(2+3.46i)T+(21.537.2i)T2 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2}
47 16T+47T2 1 - 6T + 47T^{2}
53 19T+53T2 1 - 9T + 53T^{2}
59 1+(29.551.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(2.54.33i)T+(30.552.8i)T2 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(11.73i)T+(33.5+58.0i)T2 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2}
71 1+(35.19i)T+(35.561.4i)T2 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1T+73T2 1 - T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+(35.19i)T+(44.5+77.0i)T2 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(7+12.1i)T+(48.584.0i)T2 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.35560157465230223194996454498, −9.630453088540719006615741088672, −9.209257756324542347096255573778, −8.631328607182920934063792943016, −7.09528233438551949372320990265, −5.86975554491085492382673460408, −5.53909783371407478066925571533, −4.11263979243731739002034386271, −2.97597831775266201140342160886, −2.07711158243366803850872861099, 1.10678632525517700864777588364, 2.26959389558707272582976937652, 3.36708266486775525465195576908, 4.82725718464144349283243456518, 6.05537491102806553302358037419, 6.96112979680475448652084198151, 7.32374642384608213755406093035, 8.612550988928739262052648196416, 9.406321567730820213202728551975, 10.27101224286537867897798549883

Graph of the ZZ-function along the critical line