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Properties

Label 1-1205-1205.1099-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.665 + 0.746i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + 2-s + (0.309 + 0.951i)3-s + 4-s + (0.309 + 0.951i)6-s + (−0.587 + 0.809i)7-s + 8-s + (−0.809 + 0.587i)9-s + i·11-s + (0.309 + 0.951i)12-s + (−0.951 + 0.309i)13-s + (−0.587 + 0.809i)14-s + 16-s + (0.587 − 0.809i)17-s + (−0.809 + 0.587i)18-s + i·19-s + ⋯
L(s)  = 1  + 2-s + (0.309 + 0.951i)3-s + 4-s + (0.309 + 0.951i)6-s + (−0.587 + 0.809i)7-s + 8-s + (−0.809 + 0.587i)9-s + i·11-s + (0.309 + 0.951i)12-s + (−0.951 + 0.309i)13-s + (−0.587 + 0.809i)14-s + 16-s + (0.587 − 0.809i)17-s + (−0.809 + 0.587i)18-s + i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126362562 + 2.514713327i\)
\(L(\frac12)\) \(\approx\) \(1.126362562 + 2.514713327i\)
\(L(1)\) \(\approx\) \(1.610722636 + 1.028833198i\)
\(L(1)\) \(\approx\) \(1.610722636 + 1.028833198i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + iT \)
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 \( 1 + (-0.587 - 0.809i)T \)
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 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 - iT \)
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

−20.90612970406823641854024178376, −20.034439718338749822214348368770, −19.374585445524892945967494948064, −19.11006671162746264741398046935, −17.660658873543301893165042334987, −16.97980887864796036831765331077, −16.29072531887382675260928958937, −15.20594484185824166604019723162, −14.48546564670855939152945339429, −13.80097293504448264182894253051, −13.06032088547199317213600269397, −12.68797748663614719752242946838, −11.67672189658152853322031616490, −10.93842212395031157847574148442, −10.033504456650801134080755233606, −8.83137925640663275232060364416, −7.79222008981242089091230990977, −7.15311203437657300488487979715, −6.448099003642418205494073636754, −5.63703941160754972209282430429, −4.635451623319741151419950384269, −3.28396510318585983552882033814, −3.09748985413496975934734467012, −1.79663528306893513534325098480, −0.6971830018062941856215840346, 1.87361625117213018440146821472, 2.76977713532264581601348984444, 3.37708057997977904526847008275, 4.545556833341497896515961587928, 5.03587326953709340162269205312, 5.89620180617731646386757520990, 6.93485129626955516006241458568, 7.77729179843734836039978108066, 8.96523777510999663066561091617, 9.80747208848913923987494633149, 10.3320772636579924869335716007, 11.478914044631111331006377095622, 12.24207761406130773264453832277, 12.7291069240785588524256100453, 14.003177021915628873704881414939, 14.51091776905743500410396674837, 15.20695693134014490190428328191, 15.84896156062689136196533875678, 16.532470745584825581838328492339, 17.28078757296676115752292866085, 18.61529699037599127366992259050, 19.497795764693401220429301587121, 20.06310405734733530254035783308, 21.07112770626719721424245912321, 21.291923409716044881181142383800

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