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Properties

Label 2-245-7.2-c1-0-0
Degree $2$
Conductor $245$
Sign $-0.198 - 0.980i$
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 + (−1.20 − 2.09i)3-s + (−0.5 + 0.866i)5-s + 3.41·6-s − 2.82·8-s + (−1.41 + 2.44i)9-s + (−0.707 − 1.22i)10-s + (2.91 + 5.04i)11-s + 1.58·13-s + 2.41·15-s + (2.00 − 3.46i)16-s + (2.62 + 4.54i)17-s + (−1.99 − 3.46i)18-s + (−3 + 5.19i)19-s − 8.24·22-s + (−2.29 + 3.97i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (−0.696 − 1.20i)3-s + (−0.223 + 0.387i)5-s + 1.39·6-s − 0.999·8-s + (−0.471 + 0.816i)9-s + (−0.223 − 0.387i)10-s + (0.878 + 1.52i)11-s + 0.439·13-s + 0.623·15-s + (0.500 − 0.866i)16-s + (0.635 + 1.10i)17-s + (−0.471 − 0.816i)18-s + (−0.688 + 1.19i)19-s − 1.75·22-s + (−0.478 + 0.828i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 245,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428248 + 0.523436i\)
\(L(\frac12)\) \(\approx\) \(0.428248 + 0.523436i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.91 - 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.58T + 13T^{2} \)
17 \( 1 + (-2.62 - 4.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.29 - 3.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + (0.878 + 1.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.12 + 5.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (0.621 - 1.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.12 - 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.12 + 5.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.41 - 2.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.121 + 0.210i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.82T + 71T^{2} \)
73 \( 1 + (4.24 + 7.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.74 + 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.75T + 97T^{2} \)
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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

−12.23030427274837122412270790425, −11.80505923757984909286178359368, −10.44421959877414549405559132857, −9.247967586408880602692789224880, −7.903793899955661609811754920390, −7.45099874391221177069370324412, −6.42044345362402802223658417884, −5.93858818218070980501666393336, −3.88748727199105594600408506683, −1.76591004131220072761560198239, 0.72132496680646096448762832860, 3.05216437385011477597630498379, 4.29058389914350266383639768552, 5.52474193440927636294836827764, 6.48389545464634259709186269951, 8.500521055403966488399657955050, 9.184571994332825209638063295909, 10.05931868585033517239347266719, 11.01019685893078266169375196072, 11.38882719504936330696283704415

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