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Properties

Label 2-3360-105.62-c0-0-2
Degree $2$
Conductor $3360$
Sign $0.525 + 0.850i$
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 + 1.00i·15-s + (1.41 + 1.41i)17-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 − 1.00·35-s + (1 + i)37-s − 1.41·41-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 + 1.00i·15-s + (1.41 + 1.41i)17-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 − 1.00·35-s + (1 + i)37-s − 1.41·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.525 + 0.850i$
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: \((2,\ 3360,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036190101\)
\(L(\frac12)\) \(\approx\) \(1.036190101\)
\(L(1)\) 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 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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 \( 1 + 2iT - T^{2} \)
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

−8.299542090261446822770517052834, −7.77997601204845604305511038232, −7.48864980641307776032480220716, −6.46960342381366116494048825295, −5.46862014067402229555304726118, −5.07397893881095112942983844717, −4.09291264129803258365908783732, −3.25344723551540963051324191330, −1.56896874111692806635903628188, −1.01305516357556828690413184279, 1.01967869029384785975645187616, 2.82051512212679484165847567451, 3.29859349211270150538738217531, 4.44511167527059901729445315724, 5.18028170404715425964326431655, 5.63334491364519091240691944569, 6.80362973531138789033787796906, 7.30029736544056758020901346589, 8.179160254785251011303595656304, 8.986769336380009530665503794298

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