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Properties

Label 2-245-245.153-c1-0-2
Degree $2$
Conductor $245$
Sign $-0.136 - 0.990i$
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  + (−0.0719 − 0.639i)2-s + (−1.91 + 0.670i)3-s + (1.54 − 0.353i)4-s + (−2.23 + 0.114i)5-s + (0.566 + 1.17i)6-s + (−0.171 + 2.64i)7-s + (−0.761 − 2.17i)8-s + (0.874 − 0.697i)9-s + (0.233 + 1.41i)10-s + (−1.49 + 1.87i)11-s + (−2.72 + 1.71i)12-s + (0.512 + 4.55i)13-s + (1.69 − 0.0803i)14-s + (4.20 − 1.71i)15-s + (1.52 − 0.733i)16-s + (−2.60 + 1.63i)17-s + ⋯
L(s)  = 1  + (−0.0509 − 0.451i)2-s + (−1.10 + 0.386i)3-s + (0.773 − 0.176i)4-s + (−0.998 + 0.0511i)5-s + (0.231 + 0.480i)6-s + (−0.0648 + 0.997i)7-s + (−0.269 − 0.769i)8-s + (0.291 − 0.232i)9-s + (0.0739 + 0.448i)10-s + (−0.450 + 0.565i)11-s + (−0.786 + 0.494i)12-s + (0.142 + 1.26i)13-s + (0.454 − 0.0214i)14-s + (1.08 − 0.443i)15-s + (0.380 − 0.183i)16-s + (−0.631 + 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.136 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361652 + 0.414964i\)
\(L(\frac12)\) \(\approx\) \(0.361652 + 0.414964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 - 0.114i)T \)
7 \( 1 + (0.171 - 2.64i)T \)
good2 \( 1 + (0.0719 + 0.639i)T + (-1.94 + 0.445i)T^{2} \)
3 \( 1 + (1.91 - 0.670i)T + (2.34 - 1.87i)T^{2} \)
11 \( 1 + (1.49 - 1.87i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.512 - 4.55i)T + (-12.6 + 2.89i)T^{2} \)
17 \( 1 + (2.60 - 1.63i)T + (7.37 - 15.3i)T^{2} \)
19 \( 1 + 1.45T + 19T^{2} \)
23 \( 1 + (4.24 - 6.75i)T + (-9.97 - 20.7i)T^{2} \)
29 \( 1 + (-3.44 - 0.785i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 3.91iT - 31T^{2} \)
37 \( 1 + (4.40 + 7.01i)T + (-16.0 + 33.3i)T^{2} \)
41 \( 1 + (-2.01 + 4.18i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-2.14 - 0.750i)T + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (3.40 - 0.383i)T + (45.8 - 10.4i)T^{2} \)
53 \( 1 + (-5.33 + 8.49i)T + (-22.9 - 47.7i)T^{2} \)
59 \( 1 + (-4.80 + 2.31i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (9.19 + 2.09i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (3.37 - 3.37i)T - 67iT^{2} \)
71 \( 1 + (-2.68 - 11.7i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-13.4 - 1.51i)T + (71.1 + 16.2i)T^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + (-0.146 - 0.0164i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (1.06 + 1.33i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (0.332 + 0.332i)T + 97iT^{2} \)
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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

−12.02286683623165308505524326589, −11.42434262830579150474093830834, −10.82010775351663416684269826406, −9.778007287637513044785863480788, −8.548134947690572673512290834447, −7.16631345665046894304362881315, −6.25458111272745245751818824295, −5.14895171106365289792294369531, −3.85845021255045981126552711955, −2.15916931350428340008460620717, 0.48254899836577051278045728439, 3.08100116818333192558941770915, 4.66295056393283186177347546418, 6.00202055808285699839036469593, 6.74853963105427730058684076750, 7.72785255231349979411733519575, 8.393551075404338113386654843033, 10.55636025757891557958767928120, 10.84185219824695027207884423813, 11.82558551507639724531164919759

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