↑

Properties

Label 2-245-245.202-c1-0-11
Degree $2$
Conductor $245$
Sign $0.996 - 0.0781i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.19 + 0.247i)2-s + (0.431 + 1.23i)3-s + (2.80 − 0.641i)4-s + (2.21 + 0.318i)5-s + (−1.25 − 2.59i)6-s + (−0.614 − 2.57i)7-s + (−1.83 + 0.642i)8-s + (1.01 − 0.806i)9-s + (−4.93 − 0.151i)10-s + (2.84 − 3.56i)11-s + (2.00 + 3.18i)12-s + (−2.63 + 0.297i)13-s + (1.98 + 5.49i)14-s + (0.561 + 2.86i)15-s + (−1.31 + 0.634i)16-s + (−2.81 − 4.48i)17-s + ⋯
L(s)  = 1  + (−1.55 + 0.174i)2-s + (0.249 + 0.711i)3-s + (1.40 − 0.320i)4-s + (0.989 + 0.142i)5-s + (−0.511 − 1.06i)6-s + (−0.232 − 0.972i)7-s + (−0.649 + 0.227i)8-s + (0.337 − 0.268i)9-s + (−1.56 − 0.0480i)10-s + (0.858 − 1.07i)11-s + (0.577 + 0.919i)12-s + (−0.731 + 0.0824i)13-s + (0.530 + 1.46i)14-s + (0.145 + 0.739i)15-s + (−0.329 + 0.158i)16-s + (−0.682 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.996 - 0.0781i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.996 - 0.0781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777575 + 0.0304210i\)
\(L(\frac12)\) \(\approx\) \(0.777575 + 0.0304210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.21 - 0.318i)T \)
7 \( 1 + (0.614 + 2.57i)T \)
good2 \( 1 + (2.19 - 0.247i)T + (1.94 - 0.445i)T^{2} \)
3 \( 1 + (-0.431 - 1.23i)T + (-2.34 + 1.87i)T^{2} \)
11 \( 1 + (-2.84 + 3.56i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.63 - 0.297i)T + (12.6 - 2.89i)T^{2} \)
17 \( 1 + (2.81 + 4.48i)T + (-7.37 + 15.3i)T^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + (-6.24 - 3.92i)T + (9.97 + 20.7i)T^{2} \)
29 \( 1 + (-2.84 - 0.648i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 3.79iT - 31T^{2} \)
37 \( 1 + (9.54 - 5.99i)T + (16.0 - 33.3i)T^{2} \)
41 \( 1 + (3.24 - 6.73i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.31 - 6.60i)T + (-33.6 - 26.8i)T^{2} \)
47 \( 1 + (0.542 + 4.81i)T + (-45.8 + 10.4i)T^{2} \)
53 \( 1 + (-1.46 - 0.923i)T + (22.9 + 47.7i)T^{2} \)
59 \( 1 + (-2.10 + 1.01i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.61 + 1.28i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-5.01 - 5.01i)T + 67iT^{2} \)
71 \( 1 + (-1.06 - 4.68i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.43 + 12.7i)T + (-71.1 - 16.2i)T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (-0.114 + 1.01i)T + (-80.9 - 18.4i)T^{2} \)
89 \( 1 + (3.98 + 4.99i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-4.29 + 4.29i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.54422483199234442783059737904, −10.70981726369781449675248591512, −9.781532751876709349968821694200, −9.468190199450783612419212974780, −8.591377983492406089968571666134, −7.11325331000640012875486888098, −6.59735543141383282630291146353, −4.81007580594499672735554433395, −3.17716539466329801596768302978, −1.17037624841931611305866372422, 1.65089838414279260352964043455, 2.36871643498235259444650915413, 4.96688911708025150733847448628, 6.61164143088453133209064776286, 7.14988881000787604934157814046, 8.567426707897056748854963777769, 9.043494410822582688464978990567, 9.988640512132671481304941205168, 10.69612755646884721569072760244, 12.23960657743966660821650263065

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