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Properties

Label 2-2940-105.104-c1-0-61
Degree $2$
Conductor $2940$
Sign $0.491 + 0.870i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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  + (−0.299 + 1.70i)3-s + (1.67 + 1.48i)5-s + (−2.82 − 1.02i)9-s − 6.04i·11-s − 2.23·13-s + (−3.03 + 2.40i)15-s + 2.64i·17-s − 3.11i·19-s − 3.68·23-s + (0.592 + 4.96i)25-s + (2.58 − 4.50i)27-s − 5.25i·29-s + 4.70i·31-s + (10.3 + 1.80i)33-s − 6.75i·37-s + ⋯
L(s)  = 1  + (−0.172 + 0.984i)3-s + (0.747 + 0.663i)5-s + (−0.940 − 0.340i)9-s − 1.82i·11-s − 0.620·13-s + (−0.783 + 0.621i)15-s + 0.640i·17-s − 0.713i·19-s − 0.769·23-s + (0.118 + 0.992i)25-s + (0.497 − 0.867i)27-s − 0.976i·29-s + 0.844i·31-s + (1.79 + 0.314i)33-s − 1.11i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.491 + 0.870i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.491 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.017921175\)
\(L(\frac12)\) \(\approx\) \(1.017921175\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.299 - 1.70i)T \)
5 \( 1 + (-1.67 - 1.48i)T \)
7 \( 1 \)
good11 \( 1 + 6.04iT - 11T^{2} \)
13 \( 1 + 2.23T + 13T^{2} \)
17 \( 1 - 2.64iT - 17T^{2} \)
19 \( 1 + 3.11iT - 19T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
29 \( 1 + 5.25iT - 29T^{2} \)
31 \( 1 - 4.70iT - 31T^{2} \)
37 \( 1 + 6.75iT - 37T^{2} \)
41 \( 1 + 0.179T + 41T^{2} \)
43 \( 1 + 11.7iT - 43T^{2} \)
47 \( 1 + 7.14iT - 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + 0.839iT - 71T^{2} \)
73 \( 1 + 3.80T + 73T^{2} \)
79 \( 1 + 0.107T + 79T^{2} \)
83 \( 1 + 8.52iT - 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−8.809145658890531976116795197489, −8.054716167426943347836986016654, −6.96230149670907149646419655138, −6.03270764720392338638862873924, −5.73386853336758167033126815672, −4.80468021745189740447000517327, −3.70335260714327211786015276985, −3.11176815200126990379070565403, −2.11879691784352882617900599336, −0.31386837148899230164433554372, 1.34753003603756027740107321263, 2.00304081858385254257671692518, 2.91574775405473397740028562139, 4.55840804151899155716173682795, 4.93479911533710704284801734363, 5.99129548793339151104985987467, 6.53648242851712449055448804229, 7.53617092462831388012838617731, 7.83510044581970629733328089684, 8.907037524681753333706387589686

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