↑

Properties

Label 1-105-105.32-r0-0-0
Degree $1$
Conductor $105$
Sign $0.528 + 0.848i$
Analytic cond. $0.487617$
Root an. cond. $0.487617$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + i·22-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)26-s + 29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)32-s − 34-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + i·22-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)26-s + 29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)32-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.528 + 0.848i$
Analytic conductor: \(0.487617\)
Root analytic conductor: \(0.487617\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (0:\ ),\ 0.528 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.410458732 + 0.7832199219i\)
\(L(\frac12)\) \(\approx\) \(1.410458732 + 0.7832199219i\)
\(L(1)\) \(\approx\) \(1.476677464 + 0.5508734754i\)
\(L(1)\) \(\approx\) \(1.476677464 + 0.5508734754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.45986243088741434452924010482, −28.937446343019251696345930602042, −27.65614632577086591711433770383, −26.59445275256492813903231123616, −25.08306896335298695034282979970, −24.23792121437381967965878378902, −23.29240258211898176313458610974, −22.11314081836906335726843881975, −21.44623703596392182130736102607, −20.234195207005229076373620793985, −19.32348136313820219517260261900, −18.25913594425191395036397999649, −16.5671194199174155489468749619, −15.64148429718215530826827688273, −14.18802553599217821394137154308, −13.679846001335989603256430814981, −12.14635731430072303659474224317, −11.41040042735419276322193869723, −10.132214293627130218730733273, −8.82623242480204873480352443138, −6.96354068215842024260311020616, −5.856895453491883306755942201746, −4.45809640370228121992385795147, −3.25328499630752197132259031896, −1.638806404501833589960220336230, 2.3195937116227977530828456743, 3.8658184923186593132098105955, 5.055763409223087369159679222858, 6.38405566604915437999773174333, 7.47900107379176058858088424133, 8.77536192791692769784513965787, 10.44825955166816130530406605621, 11.82702345896763232158386241644, 12.79352813170290257454875265306, 13.85625375405700939121798128639, 15.0563038951139395597528862812, 15.76751997774175786579290694964, 17.17875149364298899807977253053, 17.93413964529868432316625586786, 19.83617513541340213850209476408, 20.518222745565674022213097871012, 21.97792366401787458543062961326, 22.5295454222600922196692694296, 23.71263176298932318278132996458, 24.64921977641801101620989366933, 25.57241171385210503886618888650, 26.51954950127719280450784265811, 27.838190870733763052872113655299, 29.02943013452896599173220140468, 30.32650777316238144146570611270

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