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Properties

Label 2-245-7.4-c1-0-12
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 + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

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: $\chi_{245} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366331 - 1.19221i\)
\(L(\frac12)\) \(\approx\) \(0.366331 - 1.19221i\)
\(L(\frac{3}{2})\) 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

−11.74291919625698070796149970541, −10.78983905958577920402646224576, −9.850803394007651531624720385282, −8.697357434195908007191099584395, −8.092253227661992586510636688481, −6.60910164070938178214440645775, −6.00674978323964588713915767137, −3.54800263166815266601387412330, −2.40892251295374555899298716788, −1.22947412207451935546330306254, 2.67856263426610980927892269557, 4.18330265635333461600418901080, 5.14970475519526962756956877474, 6.78742898978664900957183333624, 7.55101801921146671748088482597, 8.923906828540431095881348715145, 9.329129697075878649319128394458, 9.970367828168895862848817401947, 11.53219108298064107691998709320, 12.36697407037730610896845093089

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