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Properties

Label 1-4235-4235.954-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.449 + 0.893i$
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  + (−0.0665 − 0.997i)2-s + (0.104 + 0.994i)3-s + (−0.991 + 0.132i)4-s + (0.985 − 0.170i)6-s + (0.198 + 0.980i)8-s + (−0.978 + 0.207i)9-s + (−0.235 − 0.971i)12-s + (−0.564 + 0.825i)13-s + (0.964 − 0.263i)16-s + (0.797 − 0.603i)17-s + (0.272 + 0.962i)18-s + (0.290 + 0.956i)19-s + (−0.0475 + 0.998i)23-s + (−0.953 + 0.299i)24-s + (0.861 + 0.508i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.0665 − 0.997i)2-s + (0.104 + 0.994i)3-s + (−0.991 + 0.132i)4-s + (0.985 − 0.170i)6-s + (0.198 + 0.980i)8-s + (−0.978 + 0.207i)9-s + (−0.235 − 0.971i)12-s + (−0.564 + 0.825i)13-s + (0.964 − 0.263i)16-s + (0.797 − 0.603i)17-s + (0.272 + 0.962i)18-s + (0.290 + 0.956i)19-s + (−0.0475 + 0.998i)23-s + (−0.953 + 0.299i)24-s + (0.861 + 0.508i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (954, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.449 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6849875053 + 1.111018073i\)
\(L(\frac12)\) \(\approx\) \(0.6849875053 + 1.111018073i\)
\(L(1)\) \(\approx\) \(0.8938577535 + 0.03871968463i\)
\(L(1)\) \(\approx\) \(0.8938577535 + 0.03871968463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0665 - 0.997i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.564 + 0.825i)T \)
17 \( 1 + (0.797 - 0.603i)T \)
19 \( 1 + (0.290 + 0.956i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.0855 - 0.996i)T \)
31 \( 1 + (-0.851 - 0.524i)T \)
37 \( 1 + (0.179 + 0.983i)T \)
41 \( 1 + (0.466 + 0.884i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.272 + 0.962i)T \)
53 \( 1 + (-0.964 - 0.263i)T \)
59 \( 1 + (-0.532 - 0.846i)T \)
61 \( 1 + (0.830 + 0.556i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (0.988 + 0.151i)T \)
79 \( 1 + (0.595 + 0.803i)T \)
83 \( 1 + (0.774 + 0.633i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (0.736 + 0.676i)T \)
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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

−17.9670612152731860568039195706, −17.35582731329993624714372755660, −16.72584427895971806253191247763, −16.03225912449403008471809685106, −15.17100209067553148974443564954, −14.46237286304238638713602359274, −14.19576016034811529278733018561, −13.18919523121052718458890015871, −12.6393229297290630445589395114, −12.30154349782308615134927927199, −11.056067587345726469686939479874, −10.40092561939578730746693229978, −9.39416741232466693704135209358, −8.80652123813313989250959529800, −8.05201209810369611576851463027, −7.4944854517293135961569520743, −6.91662155957008055809255130873, −6.1702835410260565482546250072, −5.44141560720822390338514705146, −4.891016897555401062522581822171, −3.69547039305489592497910431664, −2.96402038660852329452842316296, −1.927703843335659460655521867, −0.858277786371742217443110943349, −0.2768487667993668047220426645, 0.86164445193078790558010290689, 1.91547277012854474490178104449, 2.66504087059853696798199704122, 3.5524996016185457388223448845, 3.97790428034370238236693191017, 4.91567647308236053426117919532, 5.38480272501422617559887909897, 6.31885158039790942412590312025, 7.74091652979699468625187089895, 8.03559646009490587309140558700, 9.342734960649994868567716739753, 9.43846979775146271462918514586, 10.069105674649072984371509761071, 10.905826325627358965893228842999, 11.60983490617811955159342459402, 11.954557442956922745428020460056, 12.89495734879208011429671925053, 13.80438244836137566239167518994, 14.26299623166687988602305734133, 14.851813402960041422574409878076, 15.76531915670047562502769831112, 16.55816123083188377472099157493, 17.02837752624767149821617194790, 17.73474216289896894909022690969, 18.7156636537609968459683821713

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