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Properties

Label 2-3360-105.62-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(1.991104968\)
\(L(\frac12)\) \(\approx\) \(1.991104968\)
\(L(1)\) 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

−8.901097553089437962288413263904, −8.243043971547817568595938021979, −7.44069381078504111601238251179, −6.88332969107566686491922434992, −5.50962031112368883129233106318, −4.81486324308612423484243791029, −4.46514926603083440223673704098, −3.45182512484228446705640322786, −2.15857292081030146226679211993, −1.58184075994227475910669653191, 1.28212902969151264309184622841, 2.19899160916245033133772593712, 3.12909581580376718754805449807, 3.60120471385278982954229769164, 5.25295579043266100281683549843, 5.83819493397912759768301462543, 6.39087913118794477646735364616, 7.39112846555302918120087574074, 8.041899541064077758813907464240, 8.608956398096115462019865173768

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