Properties

Label 2-245-245.103-c1-0-7
Degree $2$
Conductor $245$
Sign $0.560 - 0.828i$
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  + (0.455 − 0.617i)2-s + (−1.13 + 0.214i)3-s + (0.415 + 1.34i)4-s + (1.85 + 1.24i)5-s + (−0.383 + 0.797i)6-s + (−2.62 + 0.358i)7-s + (2.47 + 0.864i)8-s + (−1.55 + 0.609i)9-s + (1.61 − 0.578i)10-s + (−1.26 + 3.21i)11-s + (−0.760 − 1.43i)12-s + (0.908 + 0.102i)13-s + (−0.972 + 1.78i)14-s + (−2.37 − 1.01i)15-s + (−0.672 + 0.458i)16-s + (7.82 + 0.292i)17-s + ⋯
L(s)  = 1  + (0.322 − 0.436i)2-s + (−0.654 + 0.123i)3-s + (0.207 + 0.674i)4-s + (0.830 + 0.557i)5-s + (−0.156 + 0.325i)6-s + (−0.990 + 0.135i)7-s + (0.873 + 0.305i)8-s + (−0.518 + 0.203i)9-s + (0.510 − 0.182i)10-s + (−0.380 + 0.970i)11-s + (−0.219 − 0.415i)12-s + (0.251 + 0.0283i)13-s + (−0.260 + 0.476i)14-s + (−0.612 − 0.261i)15-s + (−0.168 + 0.114i)16-s + (1.89 + 0.0710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.560 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11358 + 0.590960i\)
\(L(\frac12)\) \(\approx\) \(1.11358 + 0.590960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.85 - 1.24i)T \)
7 \( 1 + (2.62 - 0.358i)T \)
good2 \( 1 + (-0.455 + 0.617i)T + (-0.589 - 1.91i)T^{2} \)
3 \( 1 + (1.13 - 0.214i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (1.26 - 3.21i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-0.908 - 0.102i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-7.82 - 0.292i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-0.0769 + 0.133i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.280 + 7.49i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (3.21 - 0.733i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-3.14 + 1.81i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.347 + 0.183i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-1.29 - 2.69i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.05 + 5.86i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (4.96 + 3.66i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (7.82 + 4.13i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.326 - 4.35i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-3.50 + 11.3i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-8.57 - 2.29i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.76 - 7.73i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.29 - 0.954i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-11.5 - 6.69i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0998 + 0.885i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (5.52 + 14.0i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-9.86 - 9.86i)T + 97iT^{2} \)
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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

−12.36683278939415600943259612857, −11.34685779593916371956485587167, −10.36326969268884750677551853523, −9.805190195533045851206427341997, −8.282241063458381124211907876621, −7.06377511445861227796613049190, −6.10693892734183765396209811132, −5.02850701130007146429467969135, −3.40725762591535203429035176906, −2.37612842898516836129040905269, 1.05757508484757600795352321015, 3.26808001788393573433416430491, 5.24297662038173716157143240159, 5.80198357166087370672627398739, 6.40914504305900997766434926766, 7.81942881170584583855246249206, 9.295697054451407829782673285769, 10.00360255033383280315220412363, 10.94554417473956073162405323829, 12.00708611281096927175868471191

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