Properties

Label 2-3380-1.1-c1-0-26
Degree $2$
Conductor $3380$
Sign $-1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
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  − 1.80·3-s − 5-s − 1.80·7-s + 0.246·9-s + 1.55·11-s + 1.80·15-s + 0.911·17-s + 1.69·19-s + 3.24·21-s + 0.246·23-s + 25-s + 4.96·27-s − 5.29·29-s − 1.08·31-s − 2.80·33-s + 1.80·35-s + 3·37-s + 6.96·41-s + 2.26·43-s − 0.246·45-s + 8.52·47-s − 3.75·49-s − 1.64·51-s + 6.92·53-s − 1.55·55-s − 3.04·57-s − 14.7·59-s + ⋯
L(s)  = 1  − 1.04·3-s − 0.447·5-s − 0.681·7-s + 0.0823·9-s + 0.468·11-s + 0.465·15-s + 0.221·17-s + 0.388·19-s + 0.708·21-s + 0.0514·23-s + 0.200·25-s + 0.954·27-s − 0.983·29-s − 0.195·31-s − 0.487·33-s + 0.304·35-s + 0.493·37-s + 1.08·41-s + 0.345·43-s − 0.0368·45-s + 1.24·47-s − 0.536·49-s − 0.230·51-s + 0.951·53-s − 0.209·55-s − 0.403·57-s − 1.91·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + 1.80T + 3T^{2} \)
7 \( 1 + 1.80T + 7T^{2} \)
11 \( 1 - 1.55T + 11T^{2} \)
17 \( 1 - 0.911T + 17T^{2} \)
19 \( 1 - 1.69T + 19T^{2} \)
23 \( 1 - 0.246T + 23T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 + 1.08T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 6.96T + 41T^{2} \)
43 \( 1 - 2.26T + 43T^{2} \)
47 \( 1 - 8.52T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 + 9.55T + 61T^{2} \)
67 \( 1 - 5.21T + 67T^{2} \)
71 \( 1 + 5.74T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 7.75T + 79T^{2} \)
83 \( 1 - 4.94T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 9.17T + 97T^{2} \)
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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

−8.129524860351160967602615589161, −7.38935283700206309246520315499, −6.63366444798710717289948170631, −5.96264377807581054154092373009, −5.36325975673600152621247711199, −4.39573132157882191165078207005, −3.60199541478232169706213200881, −2.63832532102359650479559958733, −1.12647157484703852866612872388, 0, 1.12647157484703852866612872388, 2.63832532102359650479559958733, 3.60199541478232169706213200881, 4.39573132157882191165078207005, 5.36325975673600152621247711199, 5.96264377807581054154092373009, 6.63366444798710717289948170631, 7.38935283700206309246520315499, 8.129524860351160967602615589161

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