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Properties

Label 1-2475-2475.1102-r0-0-0
Degree $1$
Conductor $2475$
Sign $-0.939 + 0.342i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
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  + (−0.994 + 0.104i)2-s + (0.978 − 0.207i)4-s + (−0.207 + 0.978i)7-s + (−0.951 + 0.309i)8-s + (−0.207 − 0.978i)13-s + (0.104 − 0.994i)14-s + (0.913 − 0.406i)16-s + (−0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (0.207 + 0.978i)23-s + (0.309 + 0.951i)26-s + i·28-s + (0.669 + 0.743i)29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.978 − 0.207i)4-s + (−0.207 + 0.978i)7-s + (−0.951 + 0.309i)8-s + (−0.207 − 0.978i)13-s + (0.104 − 0.994i)14-s + (0.913 − 0.406i)16-s + (−0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (0.207 + 0.978i)23-s + (0.309 + 0.951i)26-s + i·28-s + (0.669 + 0.743i)29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06360162590 + 0.3596073267i\)
\(L(\frac12)\) \(\approx\) \(0.06360162590 + 0.3596073267i\)
\(L(1)\) \(\approx\) \(0.5726474389 + 0.1230939740i\)
\(L(1)\) \(\approx\) \(0.5726474389 + 0.1230939740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (-0.207 - 0.978i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.207 + 0.978i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + T \)
73 \( 1 + iT \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−19.19816030466968932788079430843, −18.47321055858734436761516103070, −17.62756156620880729176745359722, −17.13697851122854609305656149410, −16.479622558611717237733296258761, −15.80518893398238224542299004868, −15.089507250463937070257308121699, −14.10457745077279552034187760794, −13.45627805409603216999211355644, −12.522090094951739113690510722707, −11.76310656803227049076118844672, −10.95584789222986239358562674417, −10.48433291923840070574121291442, −9.65488290399671427861644049390, −8.92905033922402915759036073036, −8.323972862769020208713401229409, −7.20681048492675966848500256517, −6.88835600018105392310891255897, −6.15863614825740664232429462728, −4.79835438234571567548441649215, −4.06843461677222678504744665921, −3.02709909002498085516599999733, −2.184998149469481329134619934170, −1.232743207365673594185134097452, −0.17767571621399912464135799198, 1.16606931414897293231894284882, 2.18506737490605810763597961491, 2.84624976787817701069828485407, 3.77913805840496904073675993646, 5.3034763132770969986415550857, 5.658903401059360040727697563205, 6.63319684272401710443582241783, 7.39533455583070696904697909008, 8.19326327154817991864196240564, 8.83615394238398203613682474307, 9.54540029377847135183442560159, 10.19544517775618181900088700866, 11.044566635998209378260417742250, 11.737866591694358108213861908962, 12.43581645348369989454472892908, 13.16578848392532740897655255411, 14.31493863504115648865021465253, 15.08295329535631851450493989962, 15.63435346849690568102109552757, 16.183285438479591321239027263351, 17.08331637623830095139631583867, 17.801294476841639696242849783229, 18.314352688858251092538344699451, 18.92625925688507786751136460131, 19.808324284261280557851638401504

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