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Properties

Label 2-245-7.2-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: $\chi_{245} (226, \cdot )$
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

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

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