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Properties

Label 2-245-7.2-c1-0-3
Degree $2$
Conductor $245$
Sign $0.266 - 0.963i$
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  + (0.5 + 0.866i)3-s + (1 + 1.73i)4-s + (−0.5 + 0.866i)5-s + (1 − 1.73i)9-s + (1.5 + 2.59i)11-s + (−0.999 + 1.73i)12-s − 5·13-s − 0.999·15-s + (−1.99 + 3.46i)16-s + (1.5 + 2.59i)17-s + (1 − 1.73i)19-s − 1.99·20-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + 5·27-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.5 + 0.866i)4-s + (−0.223 + 0.387i)5-s + (0.333 − 0.577i)9-s + (0.452 + 0.783i)11-s + (−0.288 + 0.499i)12-s − 1.38·13-s − 0.258·15-s + (−0.499 + 0.866i)16-s + (0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s − 0.447·20-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + 0.962·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.266 - 0.963i$
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: \((2,\ 245,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14887 + 0.874012i\)
\(L(\frac12)\) \(\approx\) \(1.14887 + 0.874012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
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 \( 1 - T + 97T^{2} \)
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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

−12.32392450593324665056395344160, −11.46080155353028838978960464645, −10.28556129907529394333505686566, −9.484976221237438958588851238989, −8.352192050458797959516434547587, −7.24233835928023693117242248699, −6.61123773043983510273189327447, −4.69368961275903088970943567662, −3.67353999802657571057822330118, −2.46794845066265842749121387058, 1.30773830699806101421822577632, 2.81610322323524882372353158831, 4.74644062665126137294936604230, 5.71690850778224111057713378588, 7.07274899484813142048842616642, 7.70998033654721587309174344678, 9.084317711588151097027953548515, 9.952359872216849956089022052653, 11.00288659065086848832981884739, 11.88970189943156110422580026500

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