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Properties

Label 2-245-245.204-c1-0-20
Degree $2$
Conductor $245$
Sign $-0.956 + 0.292i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.82 + 1.45i)2-s + (−0.976 − 2.02i)3-s + (0.770 − 3.37i)4-s + (−1.53 − 1.62i)5-s + (4.73 + 2.28i)6-s + (2.13 + 1.56i)7-s + (1.48 + 3.07i)8-s + (−1.28 + 1.61i)9-s + (5.17 + 0.733i)10-s + (−1.61 − 2.02i)11-s + (−7.59 + 1.73i)12-s + (−1.12 + 0.898i)13-s + (−6.17 + 0.261i)14-s + (−1.79 + 4.69i)15-s + (−0.956 − 0.460i)16-s + (−4.40 + 1.00i)17-s + ⋯
L(s)  = 1  + (−1.29 + 1.03i)2-s + (−0.563 − 1.17i)3-s + (0.385 − 1.68i)4-s + (−0.686 − 0.727i)5-s + (1.93 + 0.931i)6-s + (0.807 + 0.589i)7-s + (0.524 + 1.08i)8-s + (−0.428 + 0.537i)9-s + (1.63 + 0.231i)10-s + (−0.486 − 0.610i)11-s + (−2.19 + 0.500i)12-s + (−0.312 + 0.249i)13-s + (−1.65 + 0.0699i)14-s + (−0.463 + 1.21i)15-s + (−0.239 − 0.115i)16-s + (−1.06 + 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.956 + 0.292i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0168721 - 0.112962i\)
\(L(\frac12)\) \(\approx\) \(0.0168721 - 0.112962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.53 + 1.62i)T \)
7 \( 1 + (-2.13 - 1.56i)T \)
good2 \( 1 + (1.82 - 1.45i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.976 + 2.02i)T + (-1.87 + 2.34i)T^{2} \)
11 \( 1 + (1.61 + 2.02i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.12 - 0.898i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (4.40 - 1.00i)T + (15.3 - 7.37i)T^{2} \)
19 \( 1 + 8.19T + 19T^{2} \)
23 \( 1 + (-3.06 - 0.699i)T + (20.7 + 9.97i)T^{2} \)
29 \( 1 + (0.313 + 1.37i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 + (-4.01 + 0.915i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + (8.47 - 4.08i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.07 - 4.31i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.99 + 1.58i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (11.2 + 2.57i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (-7.30 - 3.51i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (0.883 + 3.87i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.14iT - 67T^{2} \)
71 \( 1 + (-0.544 + 2.38i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (13.0 + 10.4i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 - 5.85T + 79T^{2} \)
83 \( 1 + (-8.96 - 7.14i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-5.99 + 7.51i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 18.5iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.52596073398920085567275140631, −10.80020784116076149606134243500, −9.197387301168969503284112920967, −8.363771961847459436503758990739, −7.916417749947391567370587242920, −6.80250157965676483899522701876, −6.01205923435747698721078267886, −4.75570726649069770739326223470, −1.74606534820009717621076124476, −0.14423101066733404900065998961, 2.35054046834402045731932563773, 3.93411277695387899651464115120, 4.85195062040673522150395573332, 6.90536393990289569640514536026, 7.956496679043686824192359908336, 8.883663454633429100168373380516, 10.13691276192687539668825894706, 10.59471422176848336296857249850, 11.08246256666510244055995212789, 11.84905465506585393585766928182

Graph of the $Z$-function along the critical line