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Properties

Label 1-3264-3264.3125-r0-0-0
Degree $1$
Conductor $3264$
Sign $0.0415 + 0.999i$
Analytic cond. $15.1579$
Root an. cond. $15.1579$
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  + (−0.707 − 0.707i)5-s + (−0.382 + 0.923i)7-s + 11-s + (0.923 − 0.382i)13-s + (0.923 + 0.382i)19-s + (−0.382 + 0.923i)23-s − i·25-s − i·29-s + (−0.923 − 0.382i)31-s + (0.923 − 0.382i)35-s + (−0.707 + 0.707i)37-s + (0.923 + 0.382i)41-s + (0.382 + 0.923i)43-s − 47-s + (−0.707 − 0.707i)49-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (−0.382 + 0.923i)7-s + 11-s + (0.923 − 0.382i)13-s + (0.923 + 0.382i)19-s + (−0.382 + 0.923i)23-s − i·25-s − i·29-s + (−0.923 − 0.382i)31-s + (0.923 − 0.382i)35-s + (−0.707 + 0.707i)37-s + (0.923 + 0.382i)41-s + (0.382 + 0.923i)43-s − 47-s + (−0.707 − 0.707i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0415 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0415 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $0.0415 + 0.999i$
Analytic conductor: \(15.1579\)
Root analytic conductor: \(15.1579\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3264} (3125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3264,\ (0:\ ),\ 0.0415 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8518257373 + 0.8171623709i\)
\(L(\frac12)\) \(\approx\) \(0.8518257373 + 0.8171623709i\)
\(L(1)\) \(\approx\) \(0.9397132660 + 0.1180215985i\)
\(L(1)\) \(\approx\) \(0.9397132660 + 0.1180215985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + T \)
13 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 - T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (-0.923 + 0.382i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.923 + 0.382i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−18.82381172406733169497144088730, −18.04890838431914856884079850611, −17.4009771367428612762880345831, −16.38790506121287661346558715338, −16.15667025733154293136010572392, −15.265205723328383406281276619031, −14.43445674037717813340147240957, −13.94546090756354722519390799123, −13.31456951601291763042559276921, −12.215186771791813786320673657742, −11.75057251232048640220993035256, −10.81096646616428544945684834176, −10.56100114541423707515216912915, −9.447490683089560282591076913460, −8.8744205968168912641767152683, −7.83050388508949221439685563088, −7.25371094668509399009078724191, −6.5678304074306655395549470496, −6.00204147200195272172019768108, −4.7045003613582196067615721416, −3.798083108229838595360982068283, −3.65790019345167964322618827278, −2.53350894087820476260562464975, −1.39256845438527229787772307421, −0.40186499104030185453967781476, 1.10888816008680972458016426640, 1.72021513515798723056149287687, 3.18002101639581777102700089308, 3.54433816654637505951578916833, 4.490615087073760437643021266226, 5.43760689470108958305180805189, 5.94358360621016019162999011734, 6.87968664956203492708578243503, 7.7867183899641514374994045744, 8.42228922813757356489191856263, 9.22403342301641717493575717886, 9.5238961209460268126047894382, 10.75832595213479635835014060059, 11.58746578961374849102266096055, 11.957517746502147643644864886379, 12.72985950877369201694632324316, 13.327110200816362660095024195014, 14.26446626163947914531161358312, 15.01812263941688730629481447930, 15.70575596038796545633465063942, 16.239412850256561788836904277482, 16.760848804486925728101504072386, 17.84705307960634670511365862182, 18.30722527522626954973108173236, 19.17667370021012748442418339720

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