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Properties

Label 2-245-245.29-c1-0-7
Degree $2$
Conductor $245$
Sign $0.998 + 0.0629i$
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  + (0.655 − 1.36i)2-s + (−1.90 + 0.435i)3-s + (−0.176 − 0.220i)4-s + (−1.27 + 1.83i)5-s + (−0.657 + 2.87i)6-s + (2.62 − 0.335i)7-s + (2.52 − 0.577i)8-s + (0.740 − 0.356i)9-s + (1.66 + 2.93i)10-s + (3.72 + 1.79i)11-s + (0.431 + 0.344i)12-s + (−0.574 + 1.19i)13-s + (1.26 − 3.79i)14-s + (1.62 − 4.05i)15-s + (0.998 − 4.37i)16-s + (3.02 + 2.40i)17-s + ⋯
L(s)  = 1  + (0.463 − 0.962i)2-s + (−1.10 + 0.251i)3-s + (−0.0880 − 0.110i)4-s + (−0.569 + 0.822i)5-s + (−0.268 + 1.17i)6-s + (0.991 − 0.126i)7-s + (0.894 − 0.204i)8-s + (0.246 − 0.118i)9-s + (0.527 + 0.928i)10-s + (1.12 + 0.540i)11-s + (0.124 + 0.0993i)12-s + (−0.159 + 0.330i)13-s + (0.337 − 1.01i)14-s + (0.419 − 1.04i)15-s + (0.249 − 1.09i)16-s + (0.732 + 0.584i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.998 + 0.0629i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.998 + 0.0629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24691 - 0.0393128i\)
\(L(\frac12)\) \(\approx\) \(1.24691 - 0.0393128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.27 - 1.83i)T \)
7 \( 1 + (-2.62 + 0.335i)T \)
good2 \( 1 + (-0.655 + 1.36i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (1.90 - 0.435i)T + (2.70 - 1.30i)T^{2} \)
11 \( 1 + (-3.72 - 1.79i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.574 - 1.19i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (-3.02 - 2.40i)T + (3.78 + 16.5i)T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 + (6.09 - 4.86i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-1.22 + 1.54i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 8.55T + 31T^{2} \)
37 \( 1 + (1.76 + 1.40i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 + (1.10 + 4.86i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-6.43 - 1.46i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.18 - 4.53i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.73 - 1.38i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 + (-0.454 + 1.98i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-7.05 + 8.84i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 8.42iT - 67T^{2} \)
71 \( 1 + (-0.457 - 0.573i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (6.13 + 12.7i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + (-6.15 - 12.7i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.448 - 0.215i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + 9.93iT - 97T^{2} \)
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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

−11.89838329185685510475135227459, −11.28020177119602505116985500559, −10.70564363944879893918520003029, −9.718743538692323627150626273415, −7.921807292020106557353644646047, −7.07949211097324758364599253745, −5.73282787544333805250026060345, −4.44822236987415668118453564204, −3.62971443312443799617884393165, −1.78898306382043160187863689845, 1.18593696501857889977175961697, 4.13768568287575915325004000541, 5.23364891427112786210269089472, 5.76116944866552475964320701978, 6.94645166087131032700997454311, 7.901898466021805887378851648326, 8.853299683773377049638783239977, 10.45212661253274677087991910040, 11.60596546702087484144672502571, 11.83538290757329907015314017963

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