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Properties

Label 2-245-7.4-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  + (1.28 + 2.21i)2-s + (−0.780 + 1.35i)3-s + (−2.28 + 3.95i)4-s + (−0.5 − 0.866i)5-s − 4·6-s − 6.56·8-s + (0.280 + 0.486i)9-s + (1.28 − 2.21i)10-s + (0.780 − 1.35i)11-s + (−3.56 − 6.16i)12-s + 0.438·13-s + 1.56·15-s + (−3.84 − 6.65i)16-s + (0.219 − 0.379i)17-s + (−0.719 + 1.24i)18-s + (3.56 + 6.16i)19-s + ⋯
L(s)  = 1  + (0.905 + 1.56i)2-s + (−0.450 + 0.780i)3-s + (−1.14 + 1.97i)4-s + (−0.223 − 0.387i)5-s − 1.63·6-s − 2.31·8-s + (0.0935 + 0.162i)9-s + (0.405 − 0.701i)10-s + (0.235 − 0.407i)11-s + (−1.02 − 1.78i)12-s + 0.121·13-s + 0.403·15-s + (−0.960 − 1.66i)16-s + (0.0531 − 0.0920i)17-s + (−0.169 + 0.293i)18-s + (0.817 + 1.41i)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: $\chi_{245} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0956449 - 1.50716i\)
\(L(\frac12)\) \(\approx\) \(0.0956449 - 1.50716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 - 2.21i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.780 - 1.35i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.780 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
17 \( 1 + (-0.219 + 0.379i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 - 6.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.56 + 2.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.12T + 41T^{2} \)
43 \( 1 - 0.876T + 43T^{2} \)
47 \( 1 + (-4.34 - 7.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.56 + 4.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.68 + 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.12 - 8.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-6.12 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.21 - 2.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-0.561 - 0.972i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.80T + 97T^{2} \)
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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

−12.71763544809420063682819225160, −11.98821505132205538542891641972, −10.70328312672049097871993678051, −9.488139787923895464893784245523, −8.324574994839773271548333284535, −7.55999892995816385644822055878, −6.25451350288873574601827436536, −5.43458828072412253312920863806, −4.52868721539142699128846213045, −3.62636914942042653584683146474, 1.09332266482705317129743027872, 2.60611112505228715206161648371, 3.86985856923264562564570294331, 5.07112256668144490251597298378, 6.29859060935275799739591561403, 7.35488306534872737847051428555, 9.106677079350807319824598999826, 10.08122980216535299987140169055, 11.01737143857833868925683987705, 11.91496480964107186784627027933

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