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Properties

Label 1-1205-1205.24-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.922 + 0.386i$
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  + (0.5 − 0.866i)2-s + (0.104 − 0.994i)3-s + (−0.5 − 0.866i)4-s + (−0.809 − 0.587i)6-s + (−0.669 − 0.743i)7-s − 8-s + (−0.978 − 0.207i)9-s + (−0.5 + 0.866i)11-s + (−0.913 + 0.406i)12-s + (0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.5 + 0.866i)16-s + (−0.309 + 0.951i)17-s + (−0.669 + 0.743i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.104 − 0.994i)3-s + (−0.5 − 0.866i)4-s + (−0.809 − 0.587i)6-s + (−0.669 − 0.743i)7-s − 8-s + (−0.978 − 0.207i)9-s + (−0.5 + 0.866i)11-s + (−0.913 + 0.406i)12-s + (0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.5 + 0.866i)16-s + (−0.309 + 0.951i)17-s + (−0.669 + 0.743i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1780268332 + 0.03583030841i\)
\(L(\frac12)\) \(\approx\) \(0.1780268332 + 0.03583030841i\)
\(L(1)\) \(\approx\) \(0.5979633259 - 0.6463472844i\)
\(L(1)\) \(\approx\) \(0.5979633259 - 0.6463472844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−21.28095693673227375629814702503, −20.821979952873775794634635534, −19.54205598150841735965220839211, −18.77193973492073493621204039501, −17.92920759936955766933134119043, −16.83502689304926205529565946710, −16.29593704412658284555168731249, −15.822908095037169800709812283536, −15.026666409624265575844589515033, −14.381228972817318924763928613504, −13.43942755268013285536953530703, −12.87984825138201543344862821614, −11.58820711461962777761105509262, −11.16710583677901502137588256153, −9.77198453557758195704811887743, −9.020436661773811589021976090066, −8.67998936474384799199253832044, −7.48352190808304753514218763281, −6.41100352660711859524457031903, −5.81702595083537567042140147263, −4.858543600801568966451585019280, −4.24526185551680746021475534967, −3.04680817621600188817450855141, −2.635610839823834376749994097731, −0.060553820937023576491244686256, 1.23936615039858403502417749565, 2.05987968725289171174093336063, 3.11264440322274724139261182249, 3.77772825483566563450423866531, 4.98599438905732643173852736789, 5.90649000450110764605308170641, 6.68337531276167557055956557745, 7.62981817458950603619982275808, 8.50369608915671419334832828384, 9.594859829856692907233763203356, 10.43090040691055851993832365507, 10.9673697293425988490711291847, 12.2363874699886341173495701244, 12.63399467863099304993754048754, 13.29126737540910920691374035874, 13.85090868663610160263722740912, 14.956366063940648628898806890066, 15.42583514490791773013711661144, 16.98096418061086653789507957345, 17.49503821412365024948294475311, 18.41068037888469024132306668574, 19.09013692640332699024206192798, 19.73733552965155060674004937252, 20.38779600003867959487884032830, 20.93585142341057314052483135078

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