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Properties

Label 1-35e2-1225.904-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.400 - 0.916i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.963 − 0.266i)2-s + (−0.134 + 0.990i)3-s + (0.858 − 0.512i)4-s + (0.134 + 0.990i)6-s + (0.691 − 0.722i)8-s + (−0.963 − 0.266i)9-s + (−0.963 + 0.266i)11-s + (0.393 + 0.919i)12-s + (0.0448 − 0.998i)13-s + (0.473 − 0.880i)16-s + (−0.858 − 0.512i)17-s − 18-s + (0.309 − 0.951i)19-s + (−0.858 + 0.512i)22-s + (0.393 − 0.919i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (0.963 − 0.266i)2-s + (−0.134 + 0.990i)3-s + (0.858 − 0.512i)4-s + (0.134 + 0.990i)6-s + (0.691 − 0.722i)8-s + (−0.963 − 0.266i)9-s + (−0.963 + 0.266i)11-s + (0.393 + 0.919i)12-s + (0.0448 − 0.998i)13-s + (0.473 − 0.880i)16-s + (−0.858 − 0.512i)17-s − 18-s + (0.309 − 0.951i)19-s + (−0.858 + 0.512i)22-s + (0.393 − 0.919i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (904, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.753623976 - 1.147896246i\)
\(L(\frac12)\) \(\approx\) \(1.753623976 - 1.147896246i\)
\(L(1)\) \(\approx\) \(1.569624156 - 0.1970339308i\)
\(L(1)\) \(\approx\) \(1.569624156 - 0.1970339308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.963 - 0.266i)T \)
3 \( 1 + (-0.134 + 0.990i)T \)
11 \( 1 + (-0.963 + 0.266i)T \)
13 \( 1 + (0.0448 - 0.998i)T \)
17 \( 1 + (-0.858 - 0.512i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.393 - 0.919i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.393 + 0.919i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.936 - 0.351i)T \)
53 \( 1 + (-0.858 + 0.512i)T \)
59 \( 1 + (0.473 - 0.880i)T \)
61 \( 1 + (0.753 - 0.657i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (0.0448 + 0.998i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.936 + 0.351i)T \)
89 \( 1 + (-0.0448 - 0.998i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−21.26641454480231937413083595087, −20.75147237866713707317721265653, −19.61658398188465622582137880355, −19.182899196237550652457254976073, −18.0979871951118913396524438514, −17.47374022155226679404570792926, −16.46403537933937076050983588600, −15.97843593692200852653789169044, −14.857320110347348591013315588425, −14.194970553833291339461181743847, −13.411312698509762388368080782401, −12.90085859633749243690952571328, −12.13748194898616493513619744377, −11.26721748333120655161426237281, −10.780659174952051033142002425807, −9.28572726012774649225690384542, −8.197335494470897327035041526165, −7.57832280908173967470780927247, −6.78596484937909811262133583948, −5.96865754737879141731579796018, −5.333581101612975718630158943680, −4.25033367707634465422796566410, −3.220842499703360154084795573387, −2.26365165906458391033162624012, −1.48358706930028356634540896748, 0.569953962987432940186985821736, 2.431349120573432582434554673746, 2.87879496107799129089058570203, 3.98051327856417025070785328225, 4.81801497488529992277698574553, 5.32981948517649272034812384485, 6.24274137705132392716655970252, 7.27630402377705978329097688895, 8.29523883975774662806575701818, 9.472184959960823746851057186, 10.13243623877390879962035980231, 11.08970404966511755345360113840, 11.28922422576656640147641067405, 12.62572803884092651739559305611, 13.14078177900514823727174270831, 14.05555864215509322911375215758, 14.97832692541123005155979598918, 15.50252497757721545641817553516, 15.98551220299097058516327505752, 16.976848499215524586984104946079, 17.85316495478742873436504275413, 18.82075002006279321071804715332, 19.95688941168394209080778135933, 20.47964525601011376392277375521, 20.89831008700212712529657992660

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