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Properties

Label 1-4235-4235.949-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.667 - 0.744i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.851 + 0.524i)2-s + (−0.913 − 0.406i)3-s + (0.449 + 0.893i)4-s + (−0.564 − 0.825i)6-s + (−0.0855 + 0.996i)8-s + (0.669 + 0.743i)9-s + (−0.0475 − 0.998i)12-s + (0.254 + 0.967i)13-s + (−0.595 + 0.803i)16-s + (−0.640 − 0.768i)17-s + (0.179 + 0.983i)18-s + (−0.969 + 0.244i)19-s + (−0.580 − 0.814i)23-s + (0.483 − 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (0.851 + 0.524i)2-s + (−0.913 − 0.406i)3-s + (0.449 + 0.893i)4-s + (−0.564 − 0.825i)6-s + (−0.0855 + 0.996i)8-s + (0.669 + 0.743i)9-s + (−0.0475 − 0.998i)12-s + (0.254 + 0.967i)13-s + (−0.595 + 0.803i)16-s + (−0.640 − 0.768i)17-s + (0.179 + 0.983i)18-s + (−0.969 + 0.244i)19-s + (−0.580 − 0.814i)23-s + (0.483 − 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9947597583 - 0.4440451555i\)
\(L(\frac12)\) \(\approx\) \(0.9947597583 - 0.4440451555i\)
\(L(1)\) \(\approx\) \(1.063101021 + 0.2429549087i\)
\(L(1)\) \(\approx\) \(1.063101021 + 0.2429549087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.851 + 0.524i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.254 + 0.967i)T \)
17 \( 1 + (-0.640 - 0.768i)T \)
19 \( 1 + (-0.969 + 0.244i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
31 \( 1 + (-0.935 - 0.353i)T \)
37 \( 1 + (-0.988 - 0.151i)T \)
41 \( 1 + (0.610 - 0.791i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.179 - 0.983i)T \)
53 \( 1 + (0.595 + 0.803i)T \)
59 \( 1 + (0.380 - 0.924i)T \)
61 \( 1 + (0.879 + 0.475i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (-0.861 - 0.508i)T \)
79 \( 1 + (0.123 - 0.992i)T \)
83 \( 1 + (0.736 - 0.676i)T \)
89 \( 1 + (-0.786 - 0.618i)T \)
97 \( 1 + (-0.516 - 0.856i)T \)
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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

−18.42268739642420184373435783411, −17.70969159781364077459389172303, −17.21357663145284089799757907614, −16.191035035057469735833123130542, −15.69859283248502906671993544979, −15.06860697226385859993083431797, −14.54360929660952901153845164420, −13.42627178726784903969044151830, −12.91346567854427970704845001329, −12.42964684988744555529465232393, −11.52015302618502059948580901946, −11.03945400694363315986376811224, −10.46974694515005806729643529488, −9.840640901304026987165787531421, −9.0446865870282050048608528024, −7.98095450733670211561853133415, −6.986222321540295921277841980276, −6.264393484953738552450580088103, −5.6964712242450901879098559175, −5.11802815874700860949725704664, −4.09443651575862259517931607650, −3.864985522825330227387691706284, −2.759814811442048050568072821049, −1.82738300101067987690542922465, −0.93111087440193323093249469635, 0.27194581243257728919862045923, 1.89092153182253796949536244521, 2.2196956449819165999664855393, 3.59618980683108321871172419426, 4.283240918610188943453470940926, 4.904929910661290267109668772171, 5.72776457039955323724658649139, 6.28150364696820351618302669144, 7.08332241125723114257885479905, 7.348915104107769285777566847527, 8.53651208193723724396978122231, 9.05483383620225783237504666612, 10.37539738780089817256902195542, 10.96555304228798865864507671518, 11.60223336014989217729164747124, 12.30326111698029115487693431, 12.751052261689071267121079202139, 13.58379822710765572657706480049, 14.0812165568437348869402347317, 14.8671340277521338063316824358, 15.72208046827632402180812793502, 16.30338094590278832120062910535, 16.754206343683348004968754249413, 17.438958335839749955192939186786, 18.17587225174124336449669053512

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