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Properties

Label 1-55e2-3025.2184-r1-0-0
Degree $1$
Conductor $3025$
Sign $-0.999 + 0.00675i$
Analytic cond. $325.081$
Root an. cond. $325.081$
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  + (−0.998 − 0.0570i)2-s + (0.809 + 0.587i)3-s + (0.993 + 0.113i)4-s + (−0.774 − 0.633i)6-s + (0.516 + 0.856i)7-s + (−0.985 − 0.170i)8-s + (0.309 + 0.951i)9-s + (0.736 + 0.676i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)14-s + (0.974 + 0.226i)16-s + (−0.0285 − 0.999i)17-s + (−0.254 − 0.967i)18-s + (0.0285 − 0.999i)19-s + (−0.0855 + 0.996i)21-s + ⋯
L(s)  = 1  + (−0.998 − 0.0570i)2-s + (0.809 + 0.587i)3-s + (0.993 + 0.113i)4-s + (−0.774 − 0.633i)6-s + (0.516 + 0.856i)7-s + (−0.985 − 0.170i)8-s + (0.309 + 0.951i)9-s + (0.736 + 0.676i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)14-s + (0.974 + 0.226i)16-s + (−0.0285 − 0.999i)17-s + (−0.254 − 0.967i)18-s + (0.0285 − 0.999i)19-s + (−0.0855 + 0.996i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.999 + 0.00675i$
Analytic conductor: \(325.081\)
Root analytic conductor: \(325.081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (1:\ ),\ -0.999 + 0.00675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004379029967 + 1.297383628i\)
\(L(\frac12)\) \(\approx\) \(0.004379029967 + 1.297383628i\)
\(L(1)\) \(\approx\) \(0.8739742389 + 0.3734560000i\)
\(L(1)\) \(\approx\) \(0.8739742389 + 0.3734560000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 - 0.0570i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.516 + 0.856i)T \)
13 \( 1 + (0.993 + 0.113i)T \)
17 \( 1 + (-0.0285 - 0.999i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (-0.516 + 0.856i)T \)
29 \( 1 + (0.564 + 0.825i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.198 + 0.980i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.362 + 0.931i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (-0.362 - 0.931i)T \)
61 \( 1 + (-0.774 + 0.633i)T \)
67 \( 1 + (0.998 + 0.0570i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.897 - 0.441i)T \)
83 \( 1 + (-0.921 - 0.389i)T \)
89 \( 1 + (-0.0285 - 0.999i)T \)
97 \( 1 + (0.985 + 0.170i)T \)
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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

−18.58868132798610656443901097261, −17.94356427562228817104187212117, −17.246950954721262036739448571423, −16.64063602158578795177805229913, −15.73680456656044045351260315584, −15.072391821365525523083823753322, −14.33459842104236603735022347605, −13.7472285064632315402031240193, −12.86193192616381609090434677619, −12.12411851363420345130518995284, −11.32522543032928771294027606761, −10.44398504521580918614984504175, −10.04865404854413488569656542273, −9.02297623505894799594637287573, −8.20835343311813595645700715031, −8.05827180754542952216447039193, −7.194523557591143710159207576843, −6.35749269721032457103280535167, −5.845726796506205226661510896973, −4.21142505228997827541954799326, −3.664795006357534250168240616631, −2.61023481451944257076224614352, −1.721834202552420942420214917163, −1.18741099988056461012591668218, −0.24024584442300890415809099442, 1.23212722177625190222146439901, 1.907269587509957868964514911659, 2.91408430885154733666915420988, 3.29288704665622917101418989882, 4.61947319434202981507952327112, 5.31422779514977906076121395688, 6.35213080669385731250618989955, 7.20771732725912249295269911747, 7.99384698410660552490607080907, 8.77589384137043050332491379652, 8.949335697392289879436680563528, 9.852152282186752625498805913215, 10.531169376560855593066356439414, 11.39970707465803076181122020094, 11.774047051030728766217445871275, 12.88788036076332641707410087652, 13.79787224710420767244684614878, 14.42437186043708414805166218428, 15.41142734688484158752941210355, 15.71474779668185728669343916666, 16.20625389493904356394607128204, 17.27575387168963794575923310725, 17.94444706264504079982654717052, 18.585223089602892321785618941078, 19.121412078378003420731800637094

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