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Properties

Label 1-3744-3744.515-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.990 + 0.140i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (−0.965 + 0.258i)5-s + (−0.5 − 0.866i)7-s + (−0.258 + 0.965i)11-s − 17-s + (−0.707 + 0.707i)19-s + (0.866 + 0.5i)23-s + (0.866 − 0.5i)25-s + (−0.258 + 0.965i)29-s + (0.866 + 0.5i)31-s + (0.707 + 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.5 + 0.866i)41-s + (−0.965 − 0.258i)43-s + (0.866 − 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s + (−0.5 − 0.866i)7-s + (−0.258 + 0.965i)11-s − 17-s + (−0.707 + 0.707i)19-s + (0.866 + 0.5i)23-s + (0.866 − 0.5i)25-s + (−0.258 + 0.965i)29-s + (0.866 + 0.5i)31-s + (0.707 + 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.5 + 0.866i)41-s + (−0.965 − 0.258i)43-s + (0.866 − 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.990 + 0.140i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.990 + 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04888200691 + 0.6908117572i\)
\(L(\frac12)\) \(\approx\) \(0.04888200691 + 0.6908117572i\)
\(L(1)\) \(\approx\) \(0.7349125538 + 0.1346372360i\)
\(L(1)\) \(\approx\) \(0.7349125538 + 0.1346372360i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.258 + 0.965i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.965 + 0.258i)T \)
61 \( 1 + (0.258 - 0.965i)T \)
67 \( 1 + (0.258 + 0.965i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.965 - 0.258i)T \)
89 \( 1 + T \)
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

−18.419219139155198917259103779106, −17.26296301080195861202672702020, −16.75239762227992711068677394873, −15.853594112121590193280698707672, −15.47556402031342251893749995038, −14.99583148942978391757489802978, −13.932992057347019948345610118725, −13.07934845745826260121206149963, −12.73291573540747031231881745042, −11.78984661267455109809341775411, −11.25680085873073915079349563423, −10.696939679311099346898719213395, −9.56829144526837573949465040374, −8.82823688831363926467347942796, −8.45556762652210758066513177126, −7.64435732564216656440037513667, −6.65537278311107941528551257616, −6.15521595902014123157574432123, −5.16134358787798762039674845794, −4.47772521454467833965609198748, −3.63915191398044903364821118536, −2.81509347020863473914190688602, −2.17716446853919743145828625050, −0.67603099352479041660128175351, −0.178050924023338684447571534916, 0.88288902384072566324601821157, 1.89141135260463028701647432163, 2.99307566131444121173160383418, 3.59405492115403785150289627940, 4.4974377999274601398506855074, 4.85877245782513532769343392378, 6.27131499444038990644447125880, 6.85860963324770231640052494054, 7.40498245689323402450497795667, 8.15185334074531782660448886166, 8.92374455417693475554305349725, 9.86727371277533377763786874334, 10.4715862909705951305771481079, 11.0892997152282600442519443691, 11.84160112513610519241608327675, 12.68522509253193873329637489245, 13.096646225601101867473931871564, 14.0011060792691812409929093876, 14.79614724332369628997996160876, 15.40366150712811847197201208561, 15.883498412801464279047792085629, 16.88002044836430507447396919807, 17.172651796692804575307149503000, 18.242199723787340407557717107974, 18.750234825216924851260069740652

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