Properties

Label 2-245-35.4-c1-0-10
Degree 22
Conductor 245245
Sign 0.9620.271i0.962 - 0.271i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
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  + (0.866 − 0.5i)2-s + (2.44 + 1.41i)3-s + (−0.500 + 0.866i)4-s + (−0.448 − 2.19i)5-s + 2.82·6-s + 3i·8-s + (2.49 + 4.33i)9-s + (−1.48 − 1.67i)10-s + (−2.44 + 1.41i)12-s − 4.24i·13-s + (2 − 5.99i)15-s + (0.500 + 0.866i)16-s + (−3.67 − 2.12i)17-s + (4.33 + 2.5i)18-s + (1.41 + 2.44i)19-s + (2.12 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.41 + 0.816i)3-s + (−0.250 + 0.433i)4-s + (−0.200 − 0.979i)5-s + 1.15·6-s + 1.06i·8-s + (0.833 + 1.44i)9-s + (−0.469 − 0.529i)10-s + (−0.707 + 0.408i)12-s − 1.17i·13-s + (0.516 − 1.54i)15-s + (0.125 + 0.216i)16-s + (−0.891 − 0.514i)17-s + (1.02 + 0.589i)18-s + (0.324 + 0.561i)19-s + (0.474 + 0.158i)20-s + ⋯

Functional equation

Λ(s)=(245s/2ΓC(s)L(s)=((0.9620.271i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(245s/2ΓC(s+1/2)L(s)=((0.9620.271i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5725 \cdot 7^{2}
Sign: 0.9620.271i0.962 - 0.271i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ245(214,)\chi_{245} (214, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 245, ( :1/2), 0.9620.271i)(2,\ 245,\ (\ :1/2),\ 0.962 - 0.271i)

Particular Values

L(1)L(1) \approx 2.20239+0.304163i2.20239 + 0.304163i
L(12)L(\frac12) \approx 2.20239+0.304163i2.20239 + 0.304163i
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)
bad5 1+(0.448+2.19i)T 1 + (0.448 + 2.19i)T
7 1 1
good2 1+(0.866+0.5i)T+(11.73i)T2 1 + (-0.866 + 0.5i)T + (1 - 1.73i)T^{2}
3 1+(2.441.41i)T+(1.5+2.59i)T2 1 + (-2.44 - 1.41i)T + (1.5 + 2.59i)T^{2}
11 1+(5.59.52i)T2 1 + (-5.5 - 9.52i)T^{2}
13 1+4.24iT13T2 1 + 4.24iT - 13T^{2}
17 1+(3.67+2.12i)T+(8.5+14.7i)T2 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2}
19 1+(1.412.44i)T+(9.5+16.4i)T2 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.462i)T+(11.519.9i)T2 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2}
29 1+29T2 1 + 29T^{2}
31 1+(2.82+4.89i)T+(15.526.8i)T2 1 + (-2.82 + 4.89i)T + (-15.5 - 26.8i)T^{2}
37 1+(5.19+3i)T+(18.532.0i)T2 1 + (-5.19 + 3i)T + (18.5 - 32.0i)T^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 143T2 1 - 43T^{2}
47 1+(23.540.7i)T2 1 + (23.5 - 40.7i)T^{2}
53 1+(6.92+4i)T+(26.5+45.8i)T2 1 + (6.92 + 4i)T + (26.5 + 45.8i)T^{2}
59 1+(4.247.34i)T+(29.551.0i)T2 1 + (4.24 - 7.34i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.948.57i)T+(30.5+52.8i)T2 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2}
67 1+(10.36i)T+(33.5+58.0i)T2 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2}
71 112T+71T2 1 - 12T + 71T^{2}
73 1+(11.0+6.36i)T+(36.5+63.2i)T2 1 + (11.0 + 6.36i)T + (36.5 + 63.2i)T^{2}
79 1+(610.3i)T+(39.5+68.4i)T2 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2}
83 18.48iT83T2 1 - 8.48iT - 83T^{2}
89 1+(2.123.67i)T+(44.5+77.0i)T2 1 + (-2.12 - 3.67i)T + (-44.5 + 77.0i)T^{2}
97 1+4.24iT97T2 1 + 4.24iT - 97T^{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

−12.37977353988219624470065449624, −11.33265278059940207005371497732, −9.969479765436604935495522424797, −9.169864961539175430110744365320, −8.259211887974300560475467237654, −7.79854165026498520128286149769, −5.44412810668871832723278910550, −4.40802441655419873847173313642, −3.63532866849199310705853611054, −2.46762104220079143664097961321, 1.99296762216752513172282824167, 3.35755073767409733152042992505, 4.48810055562412327469329330369, 6.43064286418123012046063774365, 6.82698723276605308888962268806, 7.994189159490589946696554177023, 9.042500187070389005217891172872, 9.922935020890104483901921803551, 11.17101314588107632546454601131, 12.41648181675204245904693330597

Graph of the ZZ-function along the critical line