Properties

Label 2-2960-1.1-c1-0-31
Degree $2$
Conductor $2960$
Sign $-1$
Analytic cond. $23.6357$
Root an. cond. $4.86165$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.08·3-s − 5-s − 1.91·7-s + 6.49·9-s − 1.79·11-s − 0.701·13-s + 3.08·15-s + 0.463·17-s + 0.120·19-s + 5.90·21-s − 0.408·23-s + 25-s − 10.7·27-s + 1.24·29-s + 2.54·31-s + 5.53·33-s + 1.91·35-s − 37-s + 2.16·39-s + 9.54·41-s + 4.56·43-s − 6.49·45-s + 7.05·47-s − 3.32·49-s − 1.42·51-s − 3.77·53-s + 1.79·55-s + ⋯
L(s)  = 1  − 1.77·3-s − 0.447·5-s − 0.724·7-s + 2.16·9-s − 0.541·11-s − 0.194·13-s + 0.795·15-s + 0.112·17-s + 0.0276·19-s + 1.28·21-s − 0.0852·23-s + 0.200·25-s − 2.07·27-s + 0.230·29-s + 0.457·31-s + 0.963·33-s + 0.323·35-s − 0.164·37-s + 0.346·39-s + 1.49·41-s + 0.695·43-s − 0.968·45-s + 1.02·47-s − 0.475·49-s − 0.200·51-s − 0.517·53-s + 0.242·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(23.6357\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
37 \( 1 + T \)
good3 \( 1 + 3.08T + 3T^{2} \)
7 \( 1 + 1.91T + 7T^{2} \)
11 \( 1 + 1.79T + 11T^{2} \)
13 \( 1 + 0.701T + 13T^{2} \)
17 \( 1 - 0.463T + 17T^{2} \)
19 \( 1 - 0.120T + 19T^{2} \)
23 \( 1 + 0.408T + 23T^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
41 \( 1 - 9.54T + 41T^{2} \)
43 \( 1 - 4.56T + 43T^{2} \)
47 \( 1 - 7.05T + 47T^{2} \)
53 \( 1 + 3.77T + 53T^{2} \)
59 \( 1 + 1.54T + 59T^{2} \)
61 \( 1 - 8.89T + 61T^{2} \)
67 \( 1 + 3.19T + 67T^{2} \)
71 \( 1 - 4.19T + 71T^{2} \)
73 \( 1 - 6.19T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 - 1.48T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.195843894467912411693494901354, −7.38140520119390447514463922463, −6.73259521533354047163298711549, −6.03309227255381808896958874976, −5.39505960282335512821869946578, −4.60622441515514835255854847045, −3.82822632350122399359903652440, −2.58805966324303704700565235936, −1.01536208037176210836416163358, 0, 1.01536208037176210836416163358, 2.58805966324303704700565235936, 3.82822632350122399359903652440, 4.60622441515514835255854847045, 5.39505960282335512821869946578, 6.03309227255381808896958874976, 6.73259521533354047163298711549, 7.38140520119390447514463922463, 8.195843894467912411693494901354

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