↑

Properties

Label 1-91-91.19-r0-0-0
Degree $1$
Conductor $91$
Sign $0.635 + 0.771i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
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 − 3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s + i·8-s + 9-s + 10-s + i·11-s + (−0.5 − 0.866i)12-s + (−0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)18-s + i·19-s + (0.866 + 0.5i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s + i·8-s + 9-s + 10-s + i·11-s + (−0.5 − 0.866i)12-s + (−0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)18-s + i·19-s + (0.866 + 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.635 + 0.771i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208319689 + 0.5702071266i\)
\(L(\frac12)\) \(\approx\) \(1.208319689 + 0.5702071266i\)
\(L(1)\) \(\approx\) \(1.291342679 + 0.4171553613i\)
\(L(1)\) \(\approx\) \(1.291342679 + 0.4171553613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 - T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + iT \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 - T \)
67 \( 1 - iT \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
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

−29.96580282058424482273301006282, −29.40350007927791600254835646730, −28.57187424725697563742571842848, −27.473473077398775437372396425077, −26.02120009365703288043246959434, −24.55190148363026866073909330398, −23.80549311652828216335197269429, −22.64028309331244296417893010024, −21.73281159605329308128803559303, −21.32663824044912890342845423457, −19.59869175701671105301324148628, −18.48335337230909801285832996896, −17.382679249481152508657800650208, −16.09515724979688265946338380239, −14.86546319300646518690940323407, −13.55849260524934986341945660399, −12.77201928021001930361653050750, −11.2185914303452777501513526816, −10.76575789787776183393946653493, −9.40340928657492605795840366284, −6.90453605793623532903223037976, −5.9818055846367046696349471612, −5.00563085451536513197990775521, −3.32099577482077270417542840663, −1.59407057607638689810514840581, 2.055913009535515398174933037341, 4.3033378209083288960204279188, 5.28300728913573750156483862019, 6.29787702485367851375557522311, 7.46465338118035849551929046357, 9.34937593190706672598321964909, 10.77464498293225190152355514751, 12.192708299270734762381647319116, 12.85891185671033370166294260462, 14.08196533973596137535801723359, 15.44830216850067569496889638075, 16.565755029972196019438663124407, 17.311211368186690017853404888989, 18.30344321525846214339252887890, 20.46762255488176119302774148583, 21.17039761732449601262815726503, 22.43665245731296745768124772660, 22.93413392932626169479830377987, 24.30388693543930430563088710175, 24.91680226801451761312862808820, 26.10649737601978379474136937346, 27.53955524601523791443175424981, 28.81137322867424076436010077807, 29.43537414289245719867036573116, 30.55520761892314312349388824916

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