Properties

Label 2-245-7.2-c1-0-1
Degree $2$
Conductor $245$
Sign $-0.991 - 0.126i$
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  + (−0.780 + 1.35i)2-s + (1.28 + 2.21i)3-s + (−0.219 − 0.379i)4-s + (−0.5 + 0.866i)5-s − 4·6-s − 2.43·8-s + (−1.78 + 3.08i)9-s + (−0.780 − 1.35i)10-s + (−1.28 − 2.21i)11-s + (0.561 − 0.972i)12-s + 4.56·13-s − 2.56·15-s + (2.34 − 4.05i)16-s + (2.28 + 3.95i)17-s + (−2.78 − 4.81i)18-s + (−0.561 + 0.972i)19-s + ⋯
L(s)  = 1  + (−0.552 + 0.956i)2-s + (0.739 + 1.28i)3-s + (−0.109 − 0.189i)4-s + (−0.223 + 0.387i)5-s − 1.63·6-s − 0.862·8-s + (−0.593 + 1.02i)9-s + (−0.246 − 0.427i)10-s + (−0.386 − 0.668i)11-s + (0.162 − 0.280i)12-s + 1.26·13-s − 0.661·15-s + (0.585 − 1.01i)16-s + (0.553 + 0.958i)17-s + (−0.655 − 1.13i)18-s + (−0.128 + 0.223i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 245,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0714596 + 1.12605i\)
\(L(\frac12)\) \(\approx\) \(0.0714596 + 1.12605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.780 - 1.35i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.28 - 2.21i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.28 + 2.21i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 + (-2.28 - 3.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.56 + 4.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.12T + 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 + (1.84 - 3.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.56 + 2.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.68 + 8.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.12 - 5.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (2.12 + 3.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.28 + 5.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (3.56 - 6.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−12.62006744429066798154165904212, −11.17378499313080896299262535080, −10.47011955307194223574827338989, −9.396008727657851714960656143303, −8.489227506168596540044939266351, −8.056450103977964157368157518370, −6.61451015253128499123595867193, −5.57207786291029461823556139101, −3.91144286210323581491708728378, −3.07563018957957872740070176019, 1.09397042824764715285815690555, 2.24856897575602246435458230120, 3.50810304108099994306239516545, 5.54530333484489846066949453816, 6.93269721305817459068678208881, 7.84076835513160051268425121000, 8.833013694295558919258981906456, 9.536763186405128613032534823351, 10.83233322406802633473248965172, 11.67929097214995122706531054297

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