↑

Properties

Label 2-245-7.2-c1-0-7
Degree $2$
Conductor $245$
Sign $0.991 + 0.126i$
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  + (1 − 1.73i)2-s + (1.5 + 2.59i)3-s + (−0.999 − 1.73i)4-s + (−0.5 + 0.866i)5-s + 6·6-s + (−3 + 5.19i)9-s + (0.999 + 1.73i)10-s + (−0.5 − 0.866i)11-s + (3.00 − 5.19i)12-s − 3·13-s − 3·15-s + (1.99 − 3.46i)16-s + (−1.5 − 2.59i)17-s + (6 + 10.3i)18-s + (3 − 5.19i)19-s + 1.99·20-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (0.866 + 1.49i)3-s + (−0.499 − 0.866i)4-s + (−0.223 + 0.387i)5-s + 2.44·6-s + (−1 + 1.73i)9-s + (0.316 + 0.547i)10-s + (−0.150 − 0.261i)11-s + (0.866 − 1.49i)12-s − 0.832·13-s − 0.774·15-s + (0.499 − 0.866i)16-s + (−0.363 − 0.630i)17-s + (1.41 + 2.44i)18-s + (0.688 − 1.19i)19-s + 0.447·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13931 - 0.135761i\)
\(L(\frac12)\) \(\approx\) \(2.13931 - 0.135761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15T + 97T^{2} \)
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

−11.80518205282698511789905777449, −11.07682448847542967839942019117, −10.28256014711089096364837300458, −9.585831038598588300180420944100, −8.573827798231678959576283494924, −7.21177849114347979778405506475, −5.06096741857964967920932525140, −4.48396450632544360451467890435, −3.20350596770247642540931548808, −2.64988853482532921135503103236, 1.80439682841877508434819750501, 3.59394641062905211757012566824, 5.13242414711524431103582956222, 6.27741063323307409923161103976, 7.20004770810884308462970841119, 7.84274556150422326450072855576, 8.544040517581890388977522647801, 9.914595807126040473686747426724, 11.74377265472613533486315070680, 12.54972539759642104367415266528

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