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Properties

Label 1-6025-6025.119-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.993 - 0.111i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.913 − 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (0.978 − 0.207i)7-s + (−0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.913 + 0.406i)11-s + (−0.669 + 0.743i)12-s + (0.978 + 0.207i)13-s + (−0.978 − 0.207i)14-s + (−0.104 + 0.994i)16-s − 17-s + (0.978 + 0.207i)18-s + (0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.104 + 0.994i)3-s + (0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (0.978 − 0.207i)7-s + (−0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.913 + 0.406i)11-s + (−0.669 + 0.743i)12-s + (0.978 + 0.207i)13-s + (−0.978 − 0.207i)14-s + (−0.104 + 0.994i)16-s − 17-s + (0.978 + 0.207i)18-s + (0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321900670 - 0.07390839249i\)
\(L(\frac12)\) \(\approx\) \(1.321900670 - 0.07390839249i\)
\(L(1)\) \(\approx\) \(0.8470618259 + 0.08422972883i\)
\(L(1)\) \(\approx\) \(0.8470618259 + 0.08422972883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 - T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−17.83355658369639926126293547183, −17.335648676548393442912660295287, −16.426099189938112631896521017, −16.035586376006062570956781247690, −14.97977231655147741932361941715, −14.491609514939997010721778396413, −13.95727054453925739799821192895, −13.27777148627344827704149897688, −12.18465369977664320697436976611, −11.74243567818729818024933963599, −11.06169119062837588974427363370, −10.62045828808600382430206527086, −9.41347186060981599124132044070, −8.82627661609445229408379043065, −8.36916207461139011173547328910, −7.75563297645876514582238436103, −7.06855796496746196219971717626, −6.34242036633338276091865790209, −5.82018486688429568606207615137, −5.12799322931601143062820534986, −3.91099413706426206334597230751, −3.03179537214325098313331989690, −1.89159020189424839320920538364, −1.606001668614154357570246919346, −0.82513823010688432842895530081, 0.54700577341796331448926267860, 1.689667395885550786535574477399, 2.178573898573273846874310417499, 3.26036729167444186923351595661, 4.0222112053494496097884091755, 4.39211769518149600781642087876, 5.414919313412444387141863405512, 6.35948839225825648728481362804, 7.00289038027360846170924103892, 8.063873880594365735041488645626, 8.38492783331732850862311905619, 9.175095640022062880680810785203, 9.6659564747812293229527982725, 10.32276785937579381887877999463, 11.246985558254017278910088300948, 11.413870559964460699163258510578, 11.91189024057864893276226193707, 13.246927787335734176047901741204, 13.69180095589756214993956241800, 14.69797900046327784843742184073, 15.170975390860120596098500195755, 15.87317730584323423633570774733, 16.44337942453342064392428555585, 17.21672628268009082308387775747, 17.56539456391016731871243544633

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