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Properties

Label 1-4235-4235.1109-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.998 + 0.0532i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.988 − 0.151i)2-s + (−0.978 + 0.207i)3-s + (0.953 + 0.299i)4-s + (0.998 − 0.0570i)6-s + (−0.897 − 0.441i)8-s + (0.913 − 0.406i)9-s + (−0.995 − 0.0950i)12-s + (0.198 − 0.980i)13-s + (0.820 + 0.572i)16-s + (−0.991 − 0.132i)17-s + (−0.964 + 0.263i)18-s + (0.432 + 0.901i)19-s + (0.327 + 0.945i)23-s + (0.969 + 0.244i)24-s + (−0.345 + 0.938i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.988 − 0.151i)2-s + (−0.978 + 0.207i)3-s + (0.953 + 0.299i)4-s + (0.998 − 0.0570i)6-s + (−0.897 − 0.441i)8-s + (0.913 − 0.406i)9-s + (−0.995 − 0.0950i)12-s + (0.198 − 0.980i)13-s + (0.820 + 0.572i)16-s + (−0.991 − 0.132i)17-s + (−0.964 + 0.263i)18-s + (0.432 + 0.901i)19-s + (0.327 + 0.945i)23-s + (0.969 + 0.244i)24-s + (−0.345 + 0.938i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.998 + 0.0532i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.998 + 0.0532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001256449922 - 0.04716729534i\)
\(L(\frac12)\) \(\approx\) \(0.001256449922 - 0.04716729534i\)
\(L(1)\) \(\approx\) \(0.4853615809 + 0.01932996733i\)
\(L(1)\) \(\approx\) \(0.4853615809 + 0.01932996733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.988 - 0.151i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.198 - 0.980i)T \)
17 \( 1 + (-0.991 - 0.132i)T \)
19 \( 1 + (0.432 + 0.901i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (-0.0285 - 0.999i)T \)
31 \( 1 + (-0.861 - 0.508i)T \)
37 \( 1 + (0.595 - 0.803i)T \)
41 \( 1 + (-0.774 + 0.633i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.964 + 0.263i)T \)
53 \( 1 + (-0.820 + 0.572i)T \)
59 \( 1 + (-0.161 + 0.986i)T \)
61 \( 1 + (0.625 + 0.780i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (0.123 + 0.992i)T \)
79 \( 1 + (-0.532 - 0.846i)T \)
83 \( 1 + (0.974 + 0.226i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (0.696 - 0.717i)T \)
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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

−18.61877499356005185715793427691, −17.78334375039870669183994803780, −17.29882182579683942307137745393, −16.58897787132477280033700356722, −16.09518094865424642952233900544, −15.500269182373215700799748531881, −14.668728401855872257705736705565, −13.79675732106214265182872389540, −12.8874223135469514956855728338, −12.25550866309607193900336935884, −11.40004576372409185401433782905, −11.0490618488359520560199011665, −10.45281197635207654210821509386, −9.48873714337145643517222131815, −9.00077339174317463412677535916, −8.17442446435137752382175421576, −7.24803808841037806846233720511, −6.64652279025385958347003096906, −6.35732630498795862506233263242, −5.193710306053168448964343037674, −4.70017311087593486556533403309, −3.52755917478418743256442974212, −2.374333612396594375453223230173, −1.68460403129230398620701081856, −0.82588160146462567101895621715, 0.01715339095443816738107940303, 0.798401781515391021317597210900, 1.604439516790047857291262355627, 2.56312049288778317983590615125, 3.57666668200286735544166830697, 4.27022289387808998279374997929, 5.60520401729937155092483465056, 5.76362184422620964634265353949, 6.791947315313430935166194248643, 7.443751512489627777162814318513, 8.079606375193828893721831004160, 9.06898523115932851434891617955, 9.66092757539170803978058822615, 10.33715282663086689458037411482, 10.95798285456883043151430361657, 11.49210507539662429825425697421, 12.175613315985853113150777127898, 12.87572955368976527261134250327, 13.56743049272479701309173528540, 14.89160092092167152966553413628, 15.421472556656841885883613680246, 15.97245980234938088913482423942, 16.654762912449852435489599626354, 17.33194811107700699475177724753, 17.76588329584249648059973524725

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