↑

Properties

Label 2-245-35.17-c1-0-7
Degree $2$
Conductor $245$
Sign $0.578 - 0.815i$
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  + (−0.133 + 0.5i)2-s + (0.5 − 0.133i)3-s + (1.5 + 0.866i)4-s + (1.86 + 1.23i)5-s + 0.267i·6-s + (−1.36 + 1.36i)8-s + (−2.36 + 1.36i)9-s + (−0.866 + 0.767i)10-s + (1.36 − 2.36i)11-s + (0.866 + 0.232i)12-s + (−2 − 2i)13-s + (1.09 + 0.366i)15-s + (1.23 + 2.13i)16-s + (−1 − 3.73i)17-s + (−0.366 − 1.36i)18-s + (0.366 + 0.633i)19-s + ⋯
L(s)  = 1  + (−0.0947 + 0.353i)2-s + (0.288 − 0.0773i)3-s + (0.750 + 0.433i)4-s + (0.834 + 0.550i)5-s + 0.109i·6-s + (−0.482 + 0.482i)8-s + (−0.788 + 0.455i)9-s + (−0.273 + 0.242i)10-s + (0.411 − 0.713i)11-s + (0.249 + 0.0669i)12-s + (−0.554 − 0.554i)13-s + (0.283 + 0.0945i)15-s + (0.308 + 0.533i)16-s + (−0.242 − 0.905i)17-s + (−0.0862 − 0.321i)18-s + (0.0839 + 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41768 + 0.732527i\)
\(L(\frac12)\) \(\approx\) \(1.41768 + 0.732527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.86 - 1.23i)T \)
7 \( 1 \)
good2 \( 1 + (0.133 - 0.5i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-0.5 + 0.133i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (1 + 3.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.366 - 0.633i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.133 + 0.0358i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-6.46 - 3.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 4.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + (-2.83 + 2.83i)T - 43iT^{2} \)
47 \( 1 + (8.83 + 2.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.83 + 6.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.09 - 7.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.33 - 0.767i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.6 - 2.86i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (-12.9 + 3.46i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.83 - 1.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.09 + 2.09i)T + 83iT^{2} \)
89 \( 1 + (0.330 + 0.571i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.92 - 5.92i)T - 97iT^{2} \)
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

−12.09764153305846473317456099502, −11.26805802128253656866594859045, −10.42499640619369446405023620188, −9.201547376408162869584491347098, −8.225338497739255309866223726014, −7.25128396601842605418829092890, −6.26303189106745977186327868055, −5.35017392651604571999591095560, −3.19843128648142824645905099243, −2.34402929176256115520639832499, 1.61174520549788935873907986488, 2.83583777330632344433760511590, 4.59663359487590538473910174108, 5.98975151675537099346997134530, 6.67722601813082860408384906143, 8.200595171564925584220776174621, 9.412721123783757448262158240975, 9.804761413160760257189877989698, 11.01351922728638060348829532745, 11.95309456707407308937991085448

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