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Properties

Label 1-75-75.62-r0-0-0
Degree $1$
Conductor $75$
Sign $0.904 - 0.425i$
Analytic cond. $0.348298$
Root an. cond. $0.348298$
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  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + i·7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 − 0.809i)22-s + (−0.951 + 0.309i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + i·7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 − 0.809i)22-s + (−0.951 + 0.309i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(0.348298\)
Root analytic conductor: \(0.348298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (0:\ ),\ 0.904 - 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.500517376 - 0.3354053717i\)
\(L(\frac12)\) \(\approx\) \(1.500517376 - 0.3354053717i\)
\(L(1)\) \(\approx\) \(1.568325357 - 0.2637720242i\)
\(L(1)\) \(\approx\) \(1.568325357 - 0.2637720242i\)

Euler product

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

−31.49021734242127749097715920059, −30.55175148281726817928042109858, −29.58241656806275935505448549415, −28.583548396427199769802884959412, −26.83509033747514243041378817791, −26.02986409408344734360583003054, −24.73064089440329404969479777403, −23.82125164417064322726257535889, −22.80441985600756051378580499354, −21.909146499077764059741811978454, −20.45067488504524048542420376773, −19.92076112186553457926712270858, −17.902962447867428201660514618846, −16.81098876813299198004577070645, −15.70488739237512564476279692926, −14.48037543471664478420051856992, −13.528387203519761674367498103075, −12.367114658649210502218074326086, −11.13915626427146169858320328057, −9.70608496656119102617946307698, −7.622888775403995865228620534878, −6.898262392985003517434294698198, −5.11948836465730031999563450626, −4.07164998331708953342475664812, −2.346783335822827705966530568462, 2.13003584549835795485031888699, 3.4990665827546994008466567251, 5.21158575009420122640494412625, 6.13425545853426539636435369620, 7.88488524950713748321538314315, 9.598685085517007843492527326539, 11.02264667588094260024054059230, 12.10896445569681296201747886452, 13.12466705398785078645358834634, 14.43880980150086408171908887480, 15.403331356096559521499977589526, 16.518839937767002240198917284961, 18.302735615967003015561630796453, 19.38726332056656576314937507763, 20.50788704266673045546670993521, 21.89894366997041236520671893601, 22.15415500748073529763906485034, 23.85651876209231772884189300550, 24.50213092292981182765683458704, 25.650415157416032969790826720534, 27.17761442626570509790979599506, 28.473244001471756911057561638071, 29.25757391613924619160652125092, 30.36227471728137211930436141970, 31.55554150856976834658471136611

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