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Properties

Label 1-264-264.5-r1-0-0
Degree $1$
Conductor $264$
Sign $-0.964 - 0.265i$
Analytic cond. $28.3707$
Root an. cond. $28.3707$
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  + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)7-s + (0.809 − 0.587i)13-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s − 23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)29-s + (−0.809 + 0.587i)31-s + (−0.809 + 0.587i)35-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)41-s − 43-s + (−0.309 − 0.951i)47-s + (−0.809 − 0.587i)49-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)7-s + (0.809 − 0.587i)13-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s − 23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)29-s + (−0.809 + 0.587i)31-s + (−0.809 + 0.587i)35-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)41-s − 43-s + (−0.309 − 0.951i)47-s + (−0.809 − 0.587i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(28.3707\)
Root analytic conductor: \(28.3707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 264,\ (1:\ ),\ -0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1157083966 - 0.8570275220i\)
\(L(\frac12)\) \(\approx\) \(0.1157083966 - 0.8570275220i\)
\(L(1)\) \(\approx\) \(0.8092899334 - 0.3045440964i\)
\(L(1)\) \(\approx\) \(0.8092899334 - 0.3045440964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−25.92465522947669645365905772305, −25.25170774356255803563650093776, −24.06687812521019185245850178774, −23.322411028059562974613772467957, −22.41939709960320030214647984824, −21.5001816462607975067609525997, −20.59094041631542498147877506969, −19.44938224485421632943393382155, −18.51918435778463352600021996115, −18.150719442616303953032216750601, −16.470412003648088173456217363194, −15.86537001496687644513909437895, −14.7436074931739410161993869511, −14.157812343838668031916022485581, −12.61593106220895954389721403221, −11.78708103242191234123036122349, −11.04134656619544844375625468207, −9.80457161529681199352584298022, −8.573903635525891945428145100881, −7.78747750901360010497057308191, −6.550569569753916423823824532785, −5.50337870800083070351188786251, −4.10883519607160259730507838704, −3.07501617772943879650899380579, −1.68301683613816254731032224734, 0.27950191818242851783137591844, 1.459827442027614094150903360094, 3.40659191716327601468838221953, 4.23816845382300639764800628698, 5.38326035838140007798066291183, 6.79808805834286442066035991877, 7.92810427039045613224350392229, 8.537472605722811697326681994819, 10.01881978466733912084222295718, 10.94932689656055621474239153705, 11.90965704216866019339294584841, 12.97000670431136685563386810265, 13.82284900376335294789893568388, 15.03784729816038083440300141521, 15.90916404700683872583021036000, 16.82603031979670860652608541652, 17.65644087714119946722149894480, 18.86412101284911789959666784971, 19.87510769347536681893760725721, 20.43220792137234802527684995280, 21.39029162115829074843793513192, 22.66221384265562725573415789761, 23.63858583591559673019014545315, 23.906103995446197104885526191896, 25.21444571199699635553455523556

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