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Properties

Label 2-245-245.169-c1-0-25
Degree $2$
Conductor $245$
Sign $-0.909 - 0.416i$
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  + (−0.689 − 1.43i)2-s + (0.0543 + 0.0124i)3-s + (−0.326 + 0.409i)4-s + (−1.54 − 1.61i)5-s + (−0.0197 − 0.0864i)6-s + (−1.15 + 2.37i)7-s + (−2.28 − 0.521i)8-s + (−2.70 − 1.30i)9-s + (−1.24 + 3.32i)10-s + (0.217 − 0.104i)11-s + (−0.0228 + 0.0182i)12-s + (−0.881 − 1.83i)13-s + (4.20 + 0.0142i)14-s + (−0.0640 − 0.107i)15-s + (1.06 + 4.65i)16-s + (5.25 − 4.18i)17-s + ⋯
L(s)  = 1  + (−0.487 − 1.01i)2-s + (0.0314 + 0.00716i)3-s + (−0.163 + 0.204i)4-s + (−0.691 − 0.722i)5-s + (−0.00805 − 0.0352i)6-s + (−0.436 + 0.899i)7-s + (−0.808 − 0.184i)8-s + (−0.900 − 0.433i)9-s + (−0.394 + 1.05i)10-s + (0.0655 − 0.0315i)11-s + (−0.00659 + 0.00526i)12-s + (−0.244 − 0.507i)13-s + (1.12 + 0.00380i)14-s + (−0.0165 − 0.0276i)15-s + (0.265 + 1.16i)16-s + (1.27 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.909 - 0.416i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0957452 + 0.439200i\)
\(L(\frac12)\) \(\approx\) \(0.0957452 + 0.439200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.54 + 1.61i)T \)
7 \( 1 + (1.15 - 2.37i)T \)
good2 \( 1 + (0.689 + 1.43i)T + (-1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.0543 - 0.0124i)T + (2.70 + 1.30i)T^{2} \)
11 \( 1 + (-0.217 + 0.104i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.881 + 1.83i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-5.25 + 4.18i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 7.81T + 19T^{2} \)
23 \( 1 + (2.27 + 1.81i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-2.61 - 3.27i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 0.378T + 31T^{2} \)
37 \( 1 + (-8.03 + 6.41i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 + (-1.90 + 8.35i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.20 - 0.502i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.47 + 3.07i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.81 + 2.24i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + (1.73 + 7.58i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (2.01 + 2.52i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 9.75iT - 67T^{2} \)
71 \( 1 + (-4.18 + 5.24i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.37 + 9.07i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + (1.43 - 2.98i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (10.7 + 5.15i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 5.91iT - 97T^{2} \)
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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

−11.65610339221149538837636550844, −10.69604663039659921440379130168, −9.548885921072279848328012070273, −8.878567037198322137416097010725, −8.057416775884876949164435464007, −6.33907857130079411070213277901, −5.30468250787656943361054081777, −3.57757003194913797804903579183, −2.46607521814821795091759082193, −0.39460923018207843999068513973, 2.90881825271922470748192999938, 4.20224212829836637022176855520, 6.07235065649869819026506555602, 6.67097831977496538888961556305, 7.934833652837720196962090558226, 8.153986467249224935615622225354, 9.669326999726648030792635202062, 10.66238230861809638412686867892, 11.57512557592884997755328218690, 12.55890298069460931964330403927

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