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Properties

Label 1-2652-2652.2243-r1-0-0
Degree $1$
Conductor $2652$
Sign $0.477 + 0.878i$
Analytic cond. $284.996$
Root an. cond. $284.996$
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  + i·5-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s − 25-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.5 + 0.866i)35-s + (0.866 − 0.5i)37-s + (0.866 − 0.5i)41-s + (−0.5 + 0.866i)43-s − i·47-s + (0.5 + 0.866i)49-s − 53-s + ⋯
L(s)  = 1  + i·5-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s − 25-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.5 + 0.866i)35-s + (0.866 − 0.5i)37-s + (0.866 − 0.5i)41-s + (−0.5 + 0.866i)43-s − i·47-s + (0.5 + 0.866i)49-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(284.996\)
Root analytic conductor: \(284.996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (1:\ ),\ 0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.559593899 + 1.522122879i\)
\(L(\frac12)\) \(\approx\) \(2.559593899 + 1.522122879i\)
\(L(1)\) \(\approx\) \(1.285611711 + 0.3055850345i\)
\(L(1)\) \(\approx\) \(1.285611711 + 0.3055850345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good5 \( 1 + iT \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.197676035498589661645729492501, −18.02403292145593140850948337931, −17.65164192773858909194171559149, −16.896298571484268698088319890807, −16.37569119869470183077211761184, −15.48851731482063135288714815494, −14.74596421665856695183908985071, −14.01270059944897349998719312735, −13.35108174227407984144220030457, −12.624248527002606571945254267383, −11.64666261538788512689305820767, −11.46735471316232200117021186971, −10.31848214946515084392268575835, −9.39075772678598210074249907131, −9.07776247685553183713028731881, −7.87730734113956890706396120117, −7.62093906011517717976447776318, −6.532396928217930037658876151249, −5.574608791293535824360564256522, −4.836393567809420545750987575659, −4.23883246205766367009476443744, −3.42997285979465851678967926584, −2.04213718706056401295939311569, −1.36461004975215361383084936232, −0.61968459784728297088319886859, 0.80027070495741818022496391527, 1.84278391977088998950865673133, 2.597537116617167139766277069656, 3.535957432723143539679306751911, 4.24883505839736865250867867073, 5.33436282891229954840336319452, 6.04775303503131141354258328911, 6.73178275627681306581278006048, 7.66338493681011374052778533452, 8.24862851116774322878600579974, 9.14635161597833470266857890394, 9.912785559893336847917036357823, 10.7429345355325504542942127025, 11.470487973568385646498771457901, 11.835906052957041878306688899963, 12.792407351750736901015964075268, 13.918166942778748545826213946622, 14.335584932816147357403423465166, 14.81460827917059442132351190061, 15.689904119263256802465981281954, 16.407174643238154335730040618336, 17.289575465611871639903510774579, 18.020991801509162052662706651993, 18.45249639225064515605526829908, 19.18529115431603589321270035380

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