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Properties

Label 1-1288-1288.797-r0-0-0
Degree $1$
Conductor $1288$
Sign $-0.915 - 0.402i$
Analytic cond. $5.98144$
Root an. cond. $5.98144$
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  + (−0.654 + 0.755i)3-s + (0.142 − 0.989i)5-s + (−0.142 − 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.415 − 0.909i)13-s + (0.654 + 0.755i)15-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.959 − 0.281i)25-s + (0.841 + 0.540i)27-s + (−0.841 + 0.540i)29-s + (0.654 + 0.755i)31-s + (0.415 − 0.909i)33-s + (−0.142 − 0.989i)37-s + (0.415 + 0.909i)39-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)3-s + (0.142 − 0.989i)5-s + (−0.142 − 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.415 − 0.909i)13-s + (0.654 + 0.755i)15-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.959 − 0.281i)25-s + (0.841 + 0.540i)27-s + (−0.841 + 0.540i)29-s + (0.654 + 0.755i)31-s + (0.415 − 0.909i)33-s + (−0.142 − 0.989i)37-s + (0.415 + 0.909i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $-0.915 - 0.402i$
Analytic conductor: \(5.98144\)
Root analytic conductor: \(5.98144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1288,\ (0:\ ),\ -0.915 - 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07258750969 - 0.3450466059i\)
\(L(\frac12)\) \(\approx\) \(0.07258750969 - 0.3450466059i\)
\(L(1)\) \(\approx\) \(0.6947435631 - 0.07241050282i\)
\(L(1)\) \(\approx\) \(0.6947435631 - 0.07241050282i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.654 + 0.755i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 - T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−21.40269760730338827677044482906, −20.78214337140587907749808270901, −19.38846190088797977538854618965, −18.806053423045790032152169078384, −18.54901375804361786113549524496, −17.6017208137538879565781147171, −16.841671208086447855281507158914, −16.17421840940327077011271040291, −15.083619581638374505457599197818, −14.40756428462811437739575217654, −13.411968787488005556160335546573, −13.05119751875834776971865003215, −11.79122209325502308061060616577, −11.40878209227765931978159088111, −10.42224980195989206546261200698, −9.97633122611725922319357663534, −8.45513124059952095890951319034, −7.81783272299106662340899115200, −6.95389389973313336630895940374, −6.16890181546716418967997028132, −5.663145062030912322562317294441, −4.4447862458442197459096139659, −3.31919277157079908716050245779, −2.30103789314077644928298405224, −1.51078218310650719260027394768, 0.15548928456872052949627003000, 1.28680468444597789789952409447, 2.72025528174884618388561380804, 3.74621867332156901741745968180, 4.74401465841076852936053311801, 5.318352158106513783766573233767, 5.93542311242847976839988386944, 7.16754585950381364703471865154, 8.19579405317968099440204318306, 8.93876906800696764320552477085, 9.81842921626441154487484808829, 10.475672947748432483277728894643, 11.2265903040262009847560795767, 12.279232657368117832027235258042, 12.75695269421338336021528295960, 13.587812511827448788780716335121, 14.77094458328267412573061387912, 15.49704003552680925626876276643, 16.171542007652848468769606699206, 16.69242643487945105838101920790, 17.7117262738986425881549459247, 17.98674962171266034992756108015, 19.22519593696618168425399993381, 20.21046898937890724874854460089, 20.85436523175994541421067011140

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