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Properties

Label 2-245-7.4-c1-0-8
Degree $2$
Conductor $245$
Sign $0.827 - 0.561i$
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  + (0.707 + 1.22i)2-s + (0.207 − 0.358i)3-s + (−0.5 − 0.866i)5-s + 0.585·6-s + 2.82·8-s + (1.41 + 2.44i)9-s + (0.707 − 1.22i)10-s + (0.0857 − 0.148i)11-s + 4.41·13-s − 0.414·15-s + (2.00 + 3.46i)16-s + (−1.62 + 2.80i)17-s + (−1.99 + 3.46i)18-s + (−3 − 5.19i)19-s + 0.242·22-s + (−3.70 − 6.42i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (0.119 − 0.207i)3-s + (−0.223 − 0.387i)5-s + 0.239·6-s + 0.999·8-s + (0.471 + 0.816i)9-s + (0.223 − 0.387i)10-s + (0.0258 − 0.0448i)11-s + 1.22·13-s − 0.106·15-s + (0.500 + 0.866i)16-s + (−0.393 + 0.681i)17-s + (−0.471 + 0.816i)18-s + (−0.688 − 1.19i)19-s + 0.0517·22-s + (−0.772 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70541 + 0.524021i\)
\(L(\frac12)\) \(\approx\) \(1.70541 + 0.524021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.707 - 1.22i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.0857 + 0.148i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + (1.62 - 2.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.70 + 6.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 + (5.12 - 8.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.12 + 1.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3.62 - 6.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.12 - 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.12 + 1.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.41 - 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.12 + 7.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
73 \( 1 + (-4.24 + 7.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.742 + 1.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−12.60673052420967656379813839092, −10.96550082369580544091022267492, −10.60925450053105934303777562425, −8.935919536074947244906925872393, −8.098120991573659252510887467637, −7.05218203393366536299239894391, −6.16613607461706971928875539249, −5.00377842686295782256391078937, −4.02831396531318427810135079711, −1.81565736786526193292900543806, 1.86785228511217656602261990802, 3.61054951758320207414868404252, 3.95613539227698334441280996983, 5.74522575030972106312453218398, 7.02984863183736363393922262172, 8.049881280374671782188865802418, 9.373927374978728312276934322953, 10.30452218173736317095846048953, 11.30004862157309997884825778153, 11.82481563617765568944131180315

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