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Properties

Label 1-6025-6025.1934-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.499 + 0.866i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.309 + 0.951i)2-s + 3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (0.951 + 0.309i)7-s + (−0.809 − 0.587i)8-s + 9-s + (0.951 − 0.309i)11-s + (−0.809 + 0.587i)12-s + (0.587 + 0.809i)13-s + i·14-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)17-s + (0.309 + 0.951i)18-s + (−0.587 + 0.809i)19-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + 3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (0.951 + 0.309i)7-s + (−0.809 − 0.587i)8-s + 9-s + (0.951 − 0.309i)11-s + (−0.809 + 0.587i)12-s + (0.587 + 0.809i)13-s + i·14-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)17-s + (0.309 + 0.951i)18-s + (−0.587 + 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.499 + 0.866i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1934, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.499 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.041981810 + 3.535411683i\)
\(L(\frac12)\) \(\approx\) \(2.041981810 + 3.535411683i\)
\(L(1)\) \(\approx\) \(1.611449787 + 1.235331510i\)
\(L(1)\) \(\approx\) \(1.611449787 + 1.235331510i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−17.66390664422866529179824213232, −17.14012155607341719755768079875, −15.90879196480692216923723379335, −15.28591143345072614425316672833, −14.60021515409375700274525461750, −14.11378413670899623479493031891, −13.72876150892952097901576026485, −12.756964305072440081344948564399, −12.440833494099849352540897307453, −11.46747366747744004343546491126, −10.87125380829031414830546185072, −10.32226714715890100389269519215, −9.43718758735156729553730408689, −8.925644911341296199951012002610, −8.36998754991490950347694358981, −7.54218727308525444142538433844, −6.82157342294369069408430355723, −5.791501148494205268420750747573, −4.78496197624960351128792098239, −4.4746077816590608156045284345, −3.59114869334764873102817648866, −2.97375168035887950639709231882, −2.25343664066557289066575583523, −1.350803808855758399807193173254, −0.896389658998208482746170246941, 1.259498116955317547292297389661, 1.74878415465884787806191121613, 2.9405041481065820892860733202, 3.75084273873674690170095310533, 4.16107903164708907588537847198, 4.95001135106395973208790071783, 5.843003412230638632236343794356, 6.52418737261343433502690389073, 7.19634670616530417710813532893, 8.04645874242249125303957784331, 8.42086428706893640936122458737, 9.02687656519672108186088901899, 9.56450622120663930412136574769, 10.59734743051593339522843924627, 11.42856155989563555544509741542, 12.28287019284728922539769190037, 12.77961277900349760082785769568, 13.7108050776994022703319524911, 14.12988090953965776748109513039, 14.753475121433775655691091244207, 14.98022127597792741813673041461, 15.89139415245362108016773830920, 16.55755007300610938623915839658, 17.12069646969260875527982587691, 17.85797790160829265922554233767

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