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Properties

Label 1-87-87.5-r1-0-0
Degree $1$
Conductor $87$
Sign $0.0833 + 0.996i$
Analytic cond. $9.34944$
Root an. cond. $9.34944$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.222 − 0.974i)5-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (0.900 + 0.433i)10-s + (0.623 + 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + 17-s + (0.900 + 0.433i)19-s + (−0.623 + 0.781i)20-s + (−0.900 + 0.433i)22-s + (0.222 + 0.974i)23-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.222 − 0.974i)5-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (0.900 + 0.433i)10-s + (0.623 + 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.222 − 0.974i)14-s + (0.623 + 0.781i)16-s + 17-s + (0.900 + 0.433i)19-s + (−0.623 + 0.781i)20-s + (−0.900 + 0.433i)22-s + (0.222 + 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0833 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0833 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.0833 + 0.996i$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ 0.0833 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9681194684 + 0.8905022645i\)
\(L(\frac12)\) \(\approx\) \(0.9681194684 + 0.8905022645i\)
\(L(1)\) \(\approx\) \(0.8566685111 + 0.4167359980i\)
\(L(1)\) \(\approx\) \(0.8566685111 + 0.4167359980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (-0.623 + 0.781i)T \)
41 \( 1 + T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + (-0.623 + 0.781i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.900 + 0.433i)T \)
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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

−29.97970246917994436345140463746, −29.32560003500730695774027521426, −28.14802041924235257331193000731, −26.924950557948294663172343211027, −26.255071995222847133707620134281, −25.14191997051667815141836418628, −23.20082425763678106905175367340, −22.53409178198073422675890780324, −21.62222531739256837144835840572, −20.34870836658231892841934371156, −19.279948821302866141267563490414, −18.51963663879309384091936890670, −17.36249420638555996194232475380, −16.08682941867964825228108828344, −14.31682584243763705812371159008, −13.50434285116375131111478059157, −12.20940444883336492163524200201, −10.87497186362288820801824141823, −10.15124372348524244879943054349, −8.88649098234636779571008372979, −7.294486530539534207361603928311, −5.793404275196831072622109931668, −3.67395229681909455007986958979, −2.90540017604993240544233158599, −0.82526001673522256413569574887, 1.26157496029690552368255135222, 3.90591653710958531089955909786, 5.35102184383736455184793185054, 6.41155240558150899371144767951, 7.8313522639844138314448227272, 9.26434383070295816102923496199, 9.69689332674239998564622249758, 11.995080744706986609328066098512, 13.10265804713773313584069946442, 14.19844193154742436140330928044, 15.61318518946571758660181589369, 16.42166521562045052326593638827, 17.31491511404049413985800180382, 18.61798113601364909704552386058, 19.64282235685026065599432032102, 21.0450368206603942188511555519, 22.39983094142114865959072622921, 23.33147418583224973152116588372, 24.47505824016770229512872822542, 25.37374373897941544613082296483, 26.030715150940449580367853225906, 27.59902471878582372667215743592, 28.24220766581954054625595816183, 29.264155959417921831522459189335, 31.06426352463631670176825485936

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