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Properties

Label 2-3360-105.83-c0-0-0
Degree $2$
Conductor $3360$
Sign $-0.850 - 0.525i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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)3-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s − 1.00i·9-s − 2i·11-s + 1.00·15-s − 1.41·19-s + 1.00·21-s + (−1 + i)23-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·31-s + (1.41 + 1.41i)33-s + 1.00i·35-s + (−1 + i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s − 1.00i·9-s − 2i·11-s + 1.00·15-s − 1.41·19-s + 1.00·21-s + (−1 + i)23-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·31-s + (1.41 + 1.41i)33-s + 1.00i·35-s + (−1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (3233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003648191629\)
\(L(\frac12)\) \(\approx\) \(0.003648191629\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 2iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.956329869889990330922501674517, −8.569921385438352128426407058361, −7.70013943181725407322914189697, −6.66137154168050306716268588861, −6.05315009483171128782277619590, −5.33879603806747154860377589541, −4.38754643103735769130362327864, −3.71629049351568692615436301164, −3.17852555076151465994330698562, −1.14504353466881640312351660644, 0.00260592241334527775899531262, 2.14244422097784531955252992557, 2.42721770012400945830104067697, 4.00008018672018261251645179101, 4.53795871700192476365173756194, 5.68634900349519436028307880440, 6.42352510420181426065746309780, 6.89892377425366363105828746896, 7.60924013226798128275837592925, 8.252805054951469454538264898604

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