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Properties

Label 1-6145-6145.1037-r0-0-0
Degree $1$
Conductor $6145$
Sign $-0.309 + 0.951i$
Analytic cond. $28.5372$
Root an. cond. $28.5372$
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  + (0.0255 + 0.999i)2-s + (−0.277 + 0.960i)3-s + (−0.998 + 0.0511i)4-s + (−0.967 − 0.253i)6-s + (−0.0817 − 0.996i)7-s + (−0.0766 − 0.997i)8-s + (−0.845 − 0.533i)9-s + (0.893 + 0.448i)11-s + (0.228 − 0.973i)12-s + (0.690 + 0.723i)13-s + (0.994 − 0.107i)14-s + (0.994 − 0.102i)16-s + (0.909 − 0.416i)17-s + (0.511 − 0.859i)18-s + (−0.820 + 0.571i)19-s + ⋯
L(s)  = 1  + (0.0255 + 0.999i)2-s + (−0.277 + 0.960i)3-s + (−0.998 + 0.0511i)4-s + (−0.967 − 0.253i)6-s + (−0.0817 − 0.996i)7-s + (−0.0766 − 0.997i)8-s + (−0.845 − 0.533i)9-s + (0.893 + 0.448i)11-s + (0.228 − 0.973i)12-s + (0.690 + 0.723i)13-s + (0.994 − 0.107i)14-s + (0.994 − 0.102i)16-s + (0.909 − 0.416i)17-s + (0.511 − 0.859i)18-s + (−0.820 + 0.571i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $-0.309 + 0.951i$
Analytic conductor: \(28.5372\)
Root analytic conductor: \(28.5372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (0:\ ),\ -0.309 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8804784725 + 1.211893864i\)
\(L(\frac12)\) \(\approx\) \(0.8804784725 + 1.211893864i\)
\(L(1)\) \(\approx\) \(0.7327822206 + 0.6142257263i\)
\(L(1)\) \(\approx\) \(0.7327822206 + 0.6142257263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (0.0255 + 0.999i)T \)
3 \( 1 + (-0.277 + 0.960i)T \)
7 \( 1 + (-0.0817 - 0.996i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (0.690 + 0.723i)T \)
17 \( 1 + (0.909 - 0.416i)T \)
19 \( 1 + (-0.820 + 0.571i)T \)
23 \( 1 + (-0.999 + 0.0102i)T \)
29 \( 1 + (0.957 + 0.287i)T \)
31 \( 1 + (0.467 - 0.884i)T \)
37 \( 1 + (0.733 + 0.679i)T \)
41 \( 1 + (0.307 + 0.951i)T \)
43 \( 1 + (-0.243 + 0.969i)T \)
47 \( 1 + (0.648 - 0.760i)T \)
53 \( 1 + (0.0409 - 0.999i)T \)
59 \( 1 + (-0.168 - 0.985i)T \)
61 \( 1 + (-0.529 + 0.848i)T \)
67 \( 1 + (-0.853 + 0.520i)T \)
71 \( 1 + (0.764 - 0.644i)T \)
73 \( 1 + (-0.789 - 0.613i)T \)
79 \( 1 + (-0.0766 + 0.997i)T \)
83 \( 1 + (0.904 - 0.425i)T \)
89 \( 1 + (-0.316 - 0.948i)T \)
97 \( 1 + (0.592 - 0.805i)T \)
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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

−17.60608027540599907642978068758, −17.22648979142415363302206673442, −16.29360226527848454339696418019, −15.4517437344985778027678254653, −14.567786049002425330831630093776, −14.01623368469305671982657349026, −13.41785378146691488441069027533, −12.642727776704828191303729170535, −12.085564882500408527255094746969, −11.913013114599086380727337997511, −10.825870479640502567023271024, −10.57361532277947946177281921414, −9.449853355218193864840676468, −8.7412698612476723649272801499, −8.35566510877246847195396636956, −7.63123816710959468047555707295, −6.41993880654686334764526201439, −5.94659039497043914002577493888, −5.42071331870792205643885813196, −4.40310915092320793339972665710, −3.53540342963945370818518721845, −2.79992862023806985325397099118, −2.16894038013319413261728233237, −1.32757876870322187383382379702, −0.670594660964857492099442675831, 0.65567521886132509725566993195, 1.56231103429109732016782002734, 3.11833759350324699531191244472, 3.832631238494413421671757650562, 4.37309501811315089886362232097, 4.74340403453545361266977449407, 5.97798917266330450641766820547, 6.25572770315689038324040942142, 6.97724424039083482344367153444, 7.90715970055691268132836908930, 8.45178344063553868507052117905, 9.29454522656184675049636574906, 9.956141909419418176352467780694, 10.19487056042745100247942332754, 11.294401054941880228758932729313, 11.88067789251076688260348318253, 12.703755608276964800187074911153, 13.65710779338225699785673102896, 14.136911726363256112099936031193, 14.66050011870406887925777966787, 15.25080073283059237011557527534, 16.15437683166205773848314023109, 16.611494548492755521630667461006, 16.80925624501018199441874485139, 17.67139827961263486991624177178

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