Properties

Label 2-6004-1.1-c1-0-89
Degree $2$
Conductor $6004$
Sign $1$
Analytic cond. $47.9421$
Root an. cond. $6.92402$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.41·3-s + 0.861·5-s + 4.65·7-s + 8.68·9-s + 1.32·11-s − 1.15·13-s + 2.94·15-s − 4.38·17-s − 19-s + 15.9·21-s − 2.98·23-s − 4.25·25-s + 19.4·27-s + 2.42·29-s − 9.30·31-s + 4.54·33-s + 4.01·35-s + 12.1·37-s − 3.94·39-s + 8.84·41-s − 5.95·43-s + 7.48·45-s − 3.44·47-s + 14.6·49-s − 15.0·51-s − 10.8·53-s + 1.14·55-s + ⋯
L(s)  = 1  + 1.97·3-s + 0.385·5-s + 1.76·7-s + 2.89·9-s + 0.400·11-s − 0.319·13-s + 0.760·15-s − 1.06·17-s − 0.229·19-s + 3.47·21-s − 0.622·23-s − 0.851·25-s + 3.73·27-s + 0.450·29-s − 1.67·31-s + 0.791·33-s + 0.678·35-s + 1.99·37-s − 0.631·39-s + 1.38·41-s − 0.907·43-s + 1.11·45-s − 0.502·47-s + 2.09·49-s − 2.10·51-s − 1.48·53-s + 0.154·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $1$
Analytic conductor: \(47.9421\)
Root analytic conductor: \(6.92402\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.939530653\)
\(L(\frac12)\) \(\approx\) \(5.939530653\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + T \)
79 \( 1 + T \)
good3 \( 1 - 3.41T + 3T^{2} \)
5 \( 1 - 0.861T + 5T^{2} \)
7 \( 1 - 4.65T + 7T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
23 \( 1 + 2.98T + 23T^{2} \)
29 \( 1 - 2.42T + 29T^{2} \)
31 \( 1 + 9.30T + 31T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 - 8.84T + 41T^{2} \)
43 \( 1 + 5.95T + 43T^{2} \)
47 \( 1 + 3.44T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 6.73T + 59T^{2} \)
61 \( 1 + 1.08T + 61T^{2} \)
67 \( 1 + 7.05T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 4.58T + 73T^{2} \)
83 \( 1 + 3.57T + 83T^{2} \)
89 \( 1 + 4.58T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−8.137361833233229577109633701440, −7.66342560539402689778148631764, −7.00056516468723427918694727217, −6.01442643426182308675581906398, −4.82879266540481792072807167005, −4.34345578701107553195939475103, −3.68504802665785978344503655533, −2.48871728560001214762100676515, −2.04471038064033263200764390121, −1.36227098193342456270264080733, 1.36227098193342456270264080733, 2.04471038064033263200764390121, 2.48871728560001214762100676515, 3.68504802665785978344503655533, 4.34345578701107553195939475103, 4.82879266540481792072807167005, 6.01442643426182308675581906398, 7.00056516468723427918694727217, 7.66342560539402689778148631764, 8.137361833233229577109633701440

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