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Properties

Label 1-1925-1925.1084-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.913 + 0.405i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (−0.309 − 0.951i)17-s + (0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)16-s + (−0.309 − 0.951i)17-s + (0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.913 + 0.405i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.913 + 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09045103823 - 0.4266312608i\)
\(L(\frac12)\) \(\approx\) \(-0.09045103823 - 0.4266312608i\)
\(L(1)\) \(\approx\) \(0.4241710304 - 0.2987990465i\)
\(L(1)\) \(\approx\) \(0.4241710304 - 0.2987990465i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 - T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.39346662485189789836243191890, −19.52768487179266419132450921672, −18.89753856266746163071594869207, −17.91935867580801722006671155670, −17.628099820369299363668263330811, −16.567095969077969365113249746703, −16.38288372093663707483483000290, −15.58612319117151898548243134312, −14.64661541198937240749513453295, −14.328580120415964069603912210119, −12.96580138320400329576327452300, −12.11687277948494616536722191294, −11.17098554155538671703146081245, −10.84689523763119692398854089830, −9.7243221757857760930864770242, −9.46557966993692394557964745578, −8.493606295972548781282985482605, −7.524827166467237062426535602857, −6.79922028136768879519412040665, −5.94053637159395129433575135475, −5.44268527607323363879953944, −4.414041480007357710474216699205, −3.61822143450139448760975368570, −2.04936224108014819168842228948, −1.215971890240934755334389397812, 0.28079565210555183474561888199, 1.01087584903188112826767018379, 2.28613236249392564496503646325, 2.78490827949422556884008238782, 4.138804730313920380165771371044, 5.01595339105462453446682007033, 6.00042527323303185813124680136, 6.91980091343344556892457800110, 7.53554707550181405547700161803, 8.22308831161081656897949139653, 9.25085106906078878980187085674, 9.979631492793871545297432466289, 10.81613321376580115358860535409, 11.371607331414599732299832103376, 12.03058674916766964509754055780, 12.91365346585354280341821812630, 13.23612658920514839739677618791, 14.38620865262327849479943261804, 15.64880519200378918401313894825, 16.10691921905313687312902050265, 17.05643608555791140307500580488, 17.49589882566193221052832283844, 18.31642408164405070613722510995, 18.58921082529944455205703313173, 19.68646164453554578680184571858

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