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Properties

Label 2-245-7.4-c1-0-10
Degree $2$
Conductor $245$
Sign $0.605 + 0.795i$
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  + (0.207 + 0.358i)2-s + (1.20 − 2.09i)3-s + (0.914 − 1.58i)4-s + (0.5 + 0.866i)5-s + 6-s + 1.58·8-s + (−1.41 − 2.44i)9-s + (−0.207 + 0.358i)10-s + (−2.41 + 4.18i)11-s + (−2.20 − 3.82i)12-s − 0.828·13-s + 2.41·15-s + (−1.49 − 2.59i)16-s + (−0.414 + 0.717i)17-s + (0.585 − 1.01i)18-s + (−1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (0.146 + 0.253i)2-s + (0.696 − 1.20i)3-s + (0.457 − 0.791i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s + 0.560·8-s + (−0.471 − 0.816i)9-s + (−0.0654 + 0.113i)10-s + (−0.727 + 1.26i)11-s + (−0.637 − 1.10i)12-s − 0.229·13-s + 0.623·15-s + (−0.374 − 0.649i)16-s + (−0.100 + 0.174i)17-s + (0.138 − 0.239i)18-s + (−0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 245,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61186 - 0.799044i\)
\(L(\frac12)\) \(\approx\) \(1.61186 - 0.799044i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 - 0.358i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + (0.414 - 0.717i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.20 - 2.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.41 + 5.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.24 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.74 + 9.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.20 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + (-2.41 + 4.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.58 + 7.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (-1.32 - 2.30i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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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

−12.20413060018901476287166319412, −10.92674488508011376009130147128, −10.09036237969301643007856706220, −8.966964074485525725425061489611, −7.48517025675993278541730687486, −7.20340464363351816553377519970, −6.10698845133830052127967041306, −4.82272855369867949501967489722, −2.66676796780779155939280597527, −1.71978474861932443693738345440, 2.56395133097160256416162763748, 3.57077839248044345646542693328, 4.58714474224785047569850091099, 5.94843241526975937592128156737, 7.61293908866609988972723830485, 8.494008524744384346938159479645, 9.261419626748204379688563065007, 10.45334776522569217144443530284, 11.05693885311455802875133805049, 12.24852523264960016225829366173

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