↑

Properties

Label 2-245-35.27-c3-0-10
Degree $2$
Conductor $245$
Sign $0.984 - 0.176i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.43 − 3.43i)2-s + (1.63 + 1.63i)3-s + 15.6i·4-s + (−10.9 − 2.43i)5-s − 11.2i·6-s + (26.2 − 26.2i)8-s − 21.6i·9-s + (29.1 + 45.8i)10-s − 36.5·11-s + (−25.5 + 25.5i)12-s + (−43.0 − 43.0i)13-s + (−13.8 − 21.8i)15-s − 55.5·16-s + (7.69 − 7.69i)17-s + (−74.4 + 74.4i)18-s + 122.·19-s + ⋯
L(s)  = 1  + (−1.21 − 1.21i)2-s + (0.314 + 0.314i)3-s + 1.95i·4-s + (−0.976 − 0.217i)5-s − 0.764i·6-s + (1.16 − 1.16i)8-s − 0.802i·9-s + (0.922 + 1.45i)10-s − 1.00·11-s + (−0.615 + 0.615i)12-s + (−0.919 − 0.919i)13-s + (−0.238 − 0.375i)15-s − 0.867·16-s + (0.109 − 0.109i)17-s + (−0.974 + 0.974i)18-s + 1.48·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.984 - 0.176i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.984 - 0.176i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.438413 + 0.0389203i\)
\(L(\frac12)\) \(\approx\) \(0.438413 + 0.0389203i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (10.9 + 2.43i)T \)
7 \( 1 \)
good2 \( 1 + (3.43 + 3.43i)T + 8iT^{2} \)
3 \( 1 + (-1.63 - 1.63i)T + 27iT^{2} \)
11 \( 1 + 36.5T + 1.33e3T^{2} \)
13 \( 1 + (43.0 + 43.0i)T + 2.19e3iT^{2} \)
17 \( 1 + (-7.69 + 7.69i)T - 4.91e3iT^{2} \)
19 \( 1 - 122.T + 6.85e3T^{2} \)
23 \( 1 + (126. - 126. i)T - 1.21e4iT^{2} \)
29 \( 1 - 230. iT - 2.43e4T^{2} \)
31 \( 1 - 137. iT - 2.97e4T^{2} \)
37 \( 1 + (-163. - 163. i)T + 5.06e4iT^{2} \)
41 \( 1 - 227. iT - 6.89e4T^{2} \)
43 \( 1 + (-151. + 151. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-237. + 237. i)T - 1.03e5iT^{2} \)
53 \( 1 + (136. - 136. i)T - 1.48e5iT^{2} \)
59 \( 1 - 7.98T + 2.05e5T^{2} \)
61 \( 1 + 156. iT - 2.26e5T^{2} \)
67 \( 1 + (-390. - 390. i)T + 3.00e5iT^{2} \)
71 \( 1 - 271.T + 3.57e5T^{2} \)
73 \( 1 + (630. + 630. i)T + 3.89e5iT^{2} \)
79 \( 1 - 898. iT - 4.93e5T^{2} \)
83 \( 1 + (614. + 614. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.07e3T + 7.04e5T^{2} \)
97 \( 1 + (-174. + 174. i)T - 9.12e5iT^{2} \)
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

−11.66926392188477824401043070234, −10.51645225067784706135999925334, −9.837715284786724325070911427736, −8.988544593996680628747925158167, −7.962538419283325857594026716815, −7.41517542258301592447590977043, −5.19076436304078536339373218503, −3.57015776590921341226459420778, −2.87541162397505409586515497819, −0.941898280565072948681708162698, 0.33507609145262342863889390511, 2.40191633300604593167384087875, 4.51824626836182573458739661939, 5.82966342137385706794104679913, 7.18456041580977679021214904542, 7.70110288389088030461554628033, 8.255914176985979781333863028069, 9.480065407948662422347364718493, 10.32607666555811694829278344263, 11.38624968217217845703631019539

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