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Properties

Label 1-2325-2325.578-r0-0-0
Degree $1$
Conductor $2325$
Sign $0.463 - 0.886i$
Analytic cond. $10.7972$
Root an. cond. $10.7972$
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  + i·2-s − 4-s + (0.207 + 0.978i)7-s − i·8-s + (0.104 + 0.994i)11-s + (−0.866 − 0.5i)13-s + (−0.978 + 0.207i)14-s + 16-s + (0.866 − 0.5i)17-s + (−0.669 + 0.743i)19-s + (−0.994 + 0.104i)22-s + (−0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.207 − 0.978i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.207 + 0.978i)7-s − i·8-s + (0.104 + 0.994i)11-s + (−0.866 − 0.5i)13-s + (−0.978 + 0.207i)14-s + 16-s + (0.866 − 0.5i)17-s + (−0.669 + 0.743i)19-s + (−0.994 + 0.104i)22-s + (−0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.207 − 0.978i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2325\)    =    \(3 \cdot 5^{2} \cdot 31\)
Sign: $0.463 - 0.886i$
Analytic conductor: \(10.7972\)
Root analytic conductor: \(10.7972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2325} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2325,\ (0:\ ),\ 0.463 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1367525835 - 0.08284356153i\)
\(L(\frac12)\) \(\approx\) \(0.1367525835 - 0.08284356153i\)
\(L(1)\) \(\approx\) \(0.6373931894 + 0.4184613792i\)
\(L(1)\) \(\approx\) \(0.6373931894 + 0.4184613792i\)

Euler product

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

−19.9046937835699553419787814800, −19.06241629836175255773603280790, −18.59702774802243147718367022127, −17.60203998724453503569454651356, −16.871655532379643667817468825664, −16.60880763877522048837666225271, −15.20656842808401302817333841227, −14.43293541491373741810863365550, −13.832618405152011388740124089501, −13.26433141309552785137146498698, −12.389153810971300997172936014140, −11.660662871173578155002298984131, −11.00672710678832462560854769650, −10.31092907413197523044551346151, −9.702116090294636314998043167901, −8.79566502567343595483073567350, −8.06629444276652029077940452111, −7.28132481323708017286033517802, −6.210160699949102378339388493304, −5.26184396905617134869052642146, −4.43737071058570492657827021358, −3.73517522271636233477526314088, −2.9918121260913993519731401134, −1.915011693135444118513859244923, −1.10816720224807705198183268605, 0.05579899531092923615936499484, 1.66816082903791714453338468455, 2.58456528415690080290262126656, 3.753609998086986325378691479454, 4.62080955301766234298134749119, 5.359796512118060084292031586321, 5.96329778534355789601982417249, 6.83945094273407283746498976670, 7.840548816801036082517836136261, 8.03672889860547612184589533890, 9.27704663367641319228549597013, 9.65742416833772596139584213760, 10.46237749157105511959805188641, 11.74975093900640426385150063878, 12.54410748163399114325176324112, 12.739561240841154035416929120513, 14.20298944484473868144133780672, 14.42402478503939961489273797543, 15.32094338495004905912424975026, 15.663998552068037811823799787866, 16.7451141664599326798351701097, 17.19062518538041367538649712188, 18.062125931312145001185648092150, 18.53061701301828119863984870390, 19.242549861967987461760909001060

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