Properties

Label 2-3380-1.1-c1-0-40
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  − 0.445·3-s + 5-s + 0.445·7-s − 2.80·9-s − 3.24·11-s − 0.445·15-s + 2.93·17-s + 3.04·19-s − 0.198·21-s − 2.80·23-s + 25-s + 2.58·27-s + 2.15·29-s − 0.939·31-s + 1.44·33-s + 0.445·35-s − 3·37-s − 4.58·41-s + 4.63·43-s − 2.80·45-s + 7.39·47-s − 6.80·49-s − 1.30·51-s − 13.0·53-s − 3.24·55-s − 1.35·57-s + 3.17·59-s + ⋯
L(s)  = 1  − 0.256·3-s + 0.447·5-s + 0.168·7-s − 0.933·9-s − 0.979·11-s − 0.114·15-s + 0.712·17-s + 0.699·19-s − 0.0432·21-s − 0.584·23-s + 0.200·25-s + 0.496·27-s + 0.400·29-s − 0.168·31-s + 0.251·33-s + 0.0752·35-s − 0.493·37-s − 0.715·41-s + 0.706·43-s − 0.417·45-s + 1.07·47-s − 0.971·49-s − 0.183·51-s − 1.79·53-s − 0.437·55-s − 0.179·57-s + 0.413·59-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 \( 1 + 0.445T + 3T^{2} \)
7 \( 1 - 0.445T + 7T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 - 3.04T + 19T^{2} \)
23 \( 1 + 2.80T + 23T^{2} \)
29 \( 1 - 2.15T + 29T^{2} \)
31 \( 1 + 0.939T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 - 4.63T + 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 - 3.17T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + 3.40T + 71T^{2} \)
73 \( 1 - 0.317T + 73T^{2} \)
79 \( 1 + 8.18T + 79T^{2} \)
83 \( 1 - 9.62T + 83T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−8.120932380730899726464100600824, −7.67136566676092620644416657765, −6.65464932004131972100472104488, −5.78427838099588684973266631367, −5.38756689776330052663480407378, −4.55154015160109677187183609995, −3.28169615236916236210787198856, −2.64905121103709793669959372772, −1.45593528063494355925412779508, 0, 1.45593528063494355925412779508, 2.64905121103709793669959372772, 3.28169615236916236210787198856, 4.55154015160109677187183609995, 5.38756689776330052663480407378, 5.78427838099588684973266631367, 6.65464932004131972100472104488, 7.67136566676092620644416657765, 8.120932380730899726464100600824

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