Properties

Label 1-104-104.59-r0-0-0
Degree $1$
Conductor $104$
Sign $-0.852 - 0.522i$
Analytic cond. $0.482973$
Root an. cond. $0.482973$
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  + (−0.5 − 0.866i)3-s i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s i·31-s + (0.866 + 0.5i)33-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s i·31-s + (0.866 + 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.852 - 0.522i$
Analytic conductor: \(0.482973\)
Root analytic conductor: \(0.482973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (0:\ ),\ -0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1560323651 - 0.5536621276i\)
\(L(\frac12)\) \(\approx\) \(0.1560323651 - 0.5536621276i\)
\(L(1)\) \(\approx\) \(0.5814863469 - 0.4018925901i\)
\(L(1)\) \(\approx\) \(0.5814863469 - 0.4018925901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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

−29.97574274234549050496287520285, −29.071839478877398782122184432576, −28.165416634253221227371492352652, −27.02364992175032501083092440332, −26.15770567866800981009505875808, −25.432196860376878880245887226665, −23.54732108351929339185727939668, −22.91440917153911113515848401987, −21.73196424752087227953853023806, −21.3111867130420794232885041588, −19.578608722957906113931708262475, −18.63828951390590166070728047368, −17.499002826513931198000213957484, −16.20664564450725413984525730468, −15.43436247094649590093397388822, −14.427771801131092353681869586906, −12.890217934224119813018656800842, −11.57761725964315416240920412144, −10.469187541449411133326447355345, −9.76327653064516038335531572822, −8.18590136067416860058057106353, −6.41837475287374512147613335455, −5.65673199873774513842900221314, −3.85474843868895714007407009780, −2.770043114085227843898850474436, 0.59036288290125460867570716603, 2.43133252982862386566356446494, 4.49784270075083894544161553131, 5.74758110102134486210934216699, 7.03892612965073740267861882848, 8.13490685170703307236557897004, 9.5905041282896414643229906693, 10.90869582437913371392014667537, 12.44753181784409319611108750494, 12.863510261365110123313360640254, 13.981239824827666908956567331628, 15.87306335334387743441610289649, 16.65453153536574798548380049213, 17.66554537191525321832197584129, 18.81615663907485363260478295487, 19.8748983133599150790360863832, 20.785871042436299784493349870705, 22.33132610907509956757701436637, 23.34773342724818233058810126783, 23.9840409349194026338015772219, 25.17463121134519193857197204388, 25.97728453780237386825233716438, 27.582047622218384172331683855306, 28.56091165894024284607059949985, 29.19003146195579568140497110229

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