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Properties

Label 1-1265-1265.802-r1-0-0
Degree $1$
Conductor $1265$
Sign $0.651 - 0.758i$
Analytic cond. $135.943$
Root an. cond. $135.943$
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  + (0.281 + 0.959i)2-s + (−0.755 + 0.654i)3-s + (−0.841 + 0.540i)4-s + (−0.841 − 0.540i)6-s + (0.909 + 0.415i)7-s + (−0.755 − 0.654i)8-s + (0.142 − 0.989i)9-s + (0.281 − 0.959i)12-s + (−0.909 + 0.415i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.540 + 0.841i)17-s + (0.989 − 0.142i)18-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.755 + 0.654i)3-s + (−0.841 + 0.540i)4-s + (−0.841 − 0.540i)6-s + (0.909 + 0.415i)7-s + (−0.755 − 0.654i)8-s + (0.142 − 0.989i)9-s + (0.281 − 0.959i)12-s + (−0.909 + 0.415i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.540 + 0.841i)17-s + (0.989 − 0.142i)18-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(135.943\)
Root analytic conductor: \(135.943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1265,\ (1:\ ),\ 0.651 - 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4994942717 + 0.2292952488i\)
\(L(\frac12)\) \(\approx\) \(-0.4994942717 + 0.2292952488i\)
\(L(1)\) \(\approx\) \(0.4566019721 + 0.6403553123i\)
\(L(1)\) \(\approx\) \(0.4566019721 + 0.6403553123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.281 + 0.959i)T \)
3 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
13 \( 1 + (-0.909 + 0.415i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (0.989 + 0.142i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.909 + 0.415i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (0.281 + 0.959i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.989 + 0.142i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−20.01332474912129911335415897691, −19.65609443617700253700267623163, −18.58234447676519152933248704716, −18.049637491124672773670701081847, −17.34720079151208810204652915444, −16.77931017517880438633350636646, −15.38018473422994314662918012869, −14.62558775322985247160646170910, −13.64859646406645273985027374129, −13.23856952493292026373854896985, −12.192598808263131297492223082907, −11.74181905349379419450807024951, −10.891553946959974537093028448784, −10.41761655645052779936038806003, −9.34366272706482075086295211086, −8.29407173987595713889464927009, −7.46497039017905497676618079079, −6.52522455622496076906191931984, −5.3927309758374491054157550569, −4.83779445684559804590216847442, −4.0598859126655140167197218759, −2.53077569323493014796813218392, −2.01167429494880144370843218666, −0.78283252527106843903099668008, −0.14564370775816109052122630103, 1.424316147808013254768765414590, 2.90472088929028338937171947304, 4.26277246438152472341220043186, 4.56989827996355585938586828030, 5.49030507665755930596648935727, 6.22665514031039912076223187419, 7.01829943826347164605524685382, 8.104043626483194830671826974990, 8.79157129809994861427722905982, 9.6629241063830824337572257531, 10.57952535445608850796928825416, 11.482419213518900756150746985424, 12.305545479674486545816199320596, 12.898049344292444172071076431135, 14.23162801091948330501922396956, 14.79442843033508278023750196655, 15.25363638606750133052241824317, 16.27956504963909671369369092331, 16.79397473600268433730197330693, 17.610857605445928045436344144749, 18.008608502479221920814143529007, 18.991877361667161847772638329233, 20.12676796192593557920899814530, 21.29922988351294518458686854793, 21.64067383464528743669266726011

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