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Properties

Label 1-2340-2340.1519-r0-0-0
Degree $1$
Conductor $2340$
Sign $0.135 + 0.990i$
Analytic cond. $10.8669$
Root an. cond. $10.8669$
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  + (0.866 + 0.5i)7-s − i·11-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.5 + 0.866i)23-s + 29-s + (0.866 + 0.5i)31-s + (−0.866 − 0.5i)37-s + (0.866 − 0.5i)41-s + (0.5 − 0.866i)43-s + (−0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s − 53-s − i·59-s + (−0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s − i·11-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.5 + 0.866i)23-s + 29-s + (0.866 + 0.5i)31-s + (−0.866 − 0.5i)37-s + (0.866 − 0.5i)41-s + (0.5 − 0.866i)43-s + (−0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s − 53-s − i·59-s + (−0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.135 + 0.990i$
Analytic conductor: \(10.8669\)
Root analytic conductor: \(10.8669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (0:\ ),\ 0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210000247 + 1.055781183i\)
\(L(\frac12)\) \(\approx\) \(1.210000247 + 1.055781183i\)
\(L(1)\) \(\approx\) \(1.105838855 + 0.2489688878i\)
\(L(1)\) \(\approx\) \(1.105838855 + 0.2489688878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 - iT \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 - T \)
59 \( 1 - iT \)
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 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
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

−19.31338371017709606206437608086, −18.91007345850708941847327749846, −17.79034947040708445776233852239, −17.38864781926441106997013180992, −16.66042601416405795319275864244, −15.880795119356938534779135414584, −15.01737961726123754885938510129, −14.43282891357033600052682128028, −13.6671759550613337468916064437, −13.037294131336597531709040658673, −12.17430071494063224719379527314, −11.171257009300172902619263920059, −10.87843523066041356398498959784, −10.08231499481511841802360436284, −8.94232159941924614004423537090, −8.3356682619139141798016948756, −7.80794318468336249858082804423, −6.551636197420059346975601422548, −6.23926258879622664888496716129, −4.91376560635868216672591473774, −4.49015927865150209504539423248, −3.48391491774177569982614647734, −2.54018633350972251839655428328, −1.56022603243904525106828011106, −0.55564168436942081710021840077, 1.181194833690507863505091632478, 2.08140716680857604606457670334, 2.78148764494447494709485459751, 4.0320986063950168886499175717, 4.761721013309998691436180682630, 5.37317730288513799768700522852, 6.40971702546005663396276066164, 7.2050473542874442112054085267, 7.93388762749343933741074156399, 8.77643584688600819550711991915, 9.39396385615126291992333574152, 10.35373871660120072926726736956, 11.00520926840235096489997576292, 11.934301548134275652754511844413, 12.32134399698771226147335754096, 13.279106199567881733817823194342, 14.14035299846255898940655112292, 14.68907805601259088669637786554, 15.547422993585956049099917034788, 15.92406275707062048951047533264, 17.22179395785157232059099960961, 17.56972046214687999703548541487, 18.20071134837590406470401538260, 19.092181633881053402563641334696, 19.69365087355563990958837592995

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