Properties

Label 2-3360-40.29-c1-0-34
Degree $2$
Conductor $3360$
Sign $0.316 + 0.948i$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−2 − i)5-s + i·7-s + 9-s − 6·13-s + (2 + i)15-s + 2i·17-s + 4i·19-s i·21-s − 4i·23-s + (3 + 4i)25-s − 27-s + 6i·29-s + 8·31-s + (1 − 2i)35-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.894 − 0.447i)5-s + 0.377i·7-s + 0.333·9-s − 1.66·13-s + (0.516 + 0.258i)15-s + 0.485i·17-s + 0.917i·19-s − 0.218i·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s − 0.192·27-s + 1.11i·29-s + 1.43·31-s + (0.169 − 0.338i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6316687074\)
\(L(\frac12)\) \(\approx\) \(0.6316687074\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (2 + i)T \)
7 \( 1 - iT \)
good11 \( 1 - 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.330930547511987971438085556120, −7.80531120093154186454057323951, −6.96567477981761374923079124435, −6.28699624342385626985278406297, −5.16868697802840566543630991686, −4.82702048847931642479359662314, −3.89895878746919370239977757652, −2.89560237545425143389815825411, −1.69732792631369941281081976029, −0.30728108029786993346128404027, 0.74833576553058149841603470573, 2.38518871903012722702557574717, 3.19149967919829601241504471599, 4.37000814100310825777023089899, 4.73957362977125156160147748847, 5.73419146841517276423694697831, 6.76772507938308963759707337596, 7.23329015721359118440342809506, 7.77841685468538486035892709426, 8.682811177184510242258009697753

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