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Properties

Label 1-245-245.83-r0-0-0
Degree $1$
Conductor $245$
Sign $-0.997 + 0.0726i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.974 + 0.222i)3-s + (−0.623 − 0.781i)4-s + (0.222 − 0.974i)6-s + (0.974 − 0.222i)8-s + (0.900 − 0.433i)9-s + (−0.900 − 0.433i)11-s + (0.781 + 0.623i)12-s + (−0.433 + 0.900i)13-s + (−0.222 + 0.974i)16-s + (0.781 + 0.623i)17-s + i·18-s + 19-s + (0.781 − 0.623i)22-s + (−0.781 + 0.623i)23-s + (−0.900 + 0.433i)24-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.974 + 0.222i)3-s + (−0.623 − 0.781i)4-s + (0.222 − 0.974i)6-s + (0.974 − 0.222i)8-s + (0.900 − 0.433i)9-s + (−0.900 − 0.433i)11-s + (0.781 + 0.623i)12-s + (−0.433 + 0.900i)13-s + (−0.222 + 0.974i)16-s + (0.781 + 0.623i)17-s + i·18-s + 19-s + (0.781 − 0.623i)22-s + (−0.781 + 0.623i)23-s + (−0.900 + 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.997 + 0.0726i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ -0.997 + 0.0726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01260021551 + 0.3463957646i\)
\(L(\frac12)\) \(\approx\) \(0.01260021551 + 0.3463957646i\)
\(L(1)\) \(\approx\) \(0.4193588208 + 0.2819421152i\)
\(L(1)\) \(\approx\) \(0.4193588208 + 0.2819421152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
3 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.433 + 0.900i)T \)
17 \( 1 + (0.781 + 0.623i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.781 - 0.623i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (0.433 - 0.900i)T \)
53 \( 1 + (-0.781 + 0.623i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.433 + 0.900i)T \)
79 \( 1 - T \)
83 \( 1 + (0.433 + 0.900i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.859224696027465211450883681118, −24.7288087685139748985056472054, −23.586191950464099695675658195642, −22.60835003317214215549794063581, −22.15795627509271190461660579749, −20.884246032385024044658024640379, −20.255279306354867155264423731896, −18.91548745356801262417996418719, −18.23940517265724497917054227770, −17.54081978444026475969101487805, −16.55637142065595134191749320212, −15.61078152014815023539566139872, −13.963175731568962195450852887628, −12.84305659732400769801979762797, −12.24160576211601942309817255825, −11.27103691276626487150697244375, −10.27676444110315562941186585407, −9.67022720636685223023693060991, −7.96682106192296843005923649755, −7.31223658327356559254261461127, −5.57540688222599267275211898815, −4.74787280174934446508029632759, −3.222708759788718861778205118569, −1.87703677358536508773508062910, −0.33585199777795474498539467399, 1.48845215116545741071329642362, 3.810990695463713862050732032374, 5.16863289946930914561089774136, 5.7403223503431414150996361267, 6.97221034613926660694325456481, 7.82518906334318489993764916493, 9.23271239168844004091087319241, 10.110162383220447371757872397436, 11.03464427483935857047340900462, 12.2257865782298660608086274981, 13.4312411113357953436247726499, 14.51912521296831441004445408689, 15.61413522462679226785697254300, 16.389809326792940640530820994614, 16.992914767574549164291344930613, 18.17487837849800376144061535612, 18.62029736465931294924262853288, 19.87865150674622533004865434911, 21.3724870309258966311592863117, 22.0724435306547243046004009982, 23.2075967315825802500473525175, 23.84138675568452455647138660081, 24.47622450582343591854752240443, 25.85281869892450336553099042003, 26.53333436313345582912544821184

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