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Properties

Label 2-3360-105.62-c0-0-4
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: $\chi_{3360} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.614008934\)
\(L(\frac12)\) \(\approx\) \(1.614008934\)
\(L(1)\) 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

−8.791977402562647954097687325570, −8.111013212562493476619958357101, −7.54612604656420448093728938810, −6.89802886413254853611975059407, −5.62981403016433621684020422534, −5.03411739495281885285845685383, −3.88846440596530211642405574198, −3.43187919129754962136777984161, −2.78498810814441376137336085160, −1.16295158285315436460594691225, 1.20395491585040030172455819834, 2.10582144829845530470220500673, 2.99367160584725233435958537431, 4.21081918363286643490336387099, 4.77633598689207806039503849643, 5.61460365937993967388164914666, 6.81741708507595499044974562239, 7.43038993472374033110510144592, 7.916145948580076126038427425126, 8.643744402340794658761453555875

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