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Properties

Label 2-245-35.9-c1-0-6
Degree $2$
Conductor $245$
Sign $-0.192 - 0.981i$
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  + (1.73 + i)2-s + (−0.866 + 0.5i)3-s + (0.999 + 1.73i)4-s + (−0.133 + 2.23i)5-s − 1.99·6-s + (−1 + 1.73i)9-s + (−2.46 + 3.73i)10-s + (1.5 + 2.59i)11-s + (−1.73 − i)12-s − i·13-s + (−1 − 1.99i)15-s + (1.99 − 3.46i)16-s + (6.06 − 3.5i)17-s + (−3.46 + 2i)18-s + (−3.99 + 1.99i)20-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (−0.499 + 0.288i)3-s + (0.499 + 0.866i)4-s + (−0.0599 + 0.998i)5-s − 0.816·6-s + (−0.333 + 0.577i)9-s + (−0.779 + 1.18i)10-s + (0.452 + 0.783i)11-s + (−0.499 − 0.288i)12-s − 0.277i·13-s + (−0.258 − 0.516i)15-s + (0.499 − 0.866i)16-s + (1.47 − 0.848i)17-s + (−0.816 + 0.471i)18-s + (−0.894 + 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21779 + 1.47965i\)
\(L(\frac12)\) \(\approx\) \(1.21779 + 1.47965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 \)
good2 \( 1 + (-1.73 - i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (-6.06 + 3.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-2.59 - 1.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (5.19 - 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−12.25667456593887140844272947735, −11.78524177165185347572722775507, −10.44800164348138997565316391761, −9.845440192608577784313969202813, −7.958322602654503758349066311353, −7.06420490939816165034587452602, −6.11183705253241068180499896124, −5.22446732681375830146565892445, −4.15667595737226864903522379517, −2.84776177164963043991510511993, 1.35456902048582223901306731881, 3.32947055006211952107934580714, 4.30953189749034732287028842577, 5.63230401939172853906244468216, 6.09021469497093085501832378637, 7.941213867492753863456967777869, 8.930239767891022381147587954187, 10.20957305055826878498992636900, 11.46059916346629696495410173263, 12.02722302065604328652484721483

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