Properties

Label 2-3380-1.1-c1-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.79·3-s − 5-s − 1.22·7-s + 4.80·9-s − 3.73·11-s − 2.79·15-s + 4.95·17-s + 3.06·19-s − 3.43·21-s − 1.04·23-s + 25-s + 5.05·27-s + 9.02·29-s + 6.57·31-s − 10.4·33-s + 1.22·35-s + 0.619·37-s + 8.41·41-s − 8.50·43-s − 4.80·45-s + 5.65·47-s − 5.48·49-s + 13.8·51-s + 9.83·53-s + 3.73·55-s + 8.56·57-s − 12.1·59-s + ⋯
L(s)  = 1  + 1.61·3-s − 0.447·5-s − 0.464·7-s + 1.60·9-s − 1.12·11-s − 0.721·15-s + 1.20·17-s + 0.703·19-s − 0.749·21-s − 0.217·23-s + 0.200·25-s + 0.973·27-s + 1.67·29-s + 1.18·31-s − 1.81·33-s + 0.207·35-s + 0.101·37-s + 1.31·41-s − 1.29·43-s − 0.717·45-s + 0.825·47-s − 0.784·49-s + 1.93·51-s + 1.35·53-s + 0.503·55-s + 1.13·57-s − 1.58·59-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: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.017070912\)
\(L(\frac12)\) \(\approx\) \(3.017070912\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 3T^{2} \)
7 \( 1 + 1.22T + 7T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
17 \( 1 - 4.95T + 17T^{2} \)
19 \( 1 - 3.06T + 19T^{2} \)
23 \( 1 + 1.04T + 23T^{2} \)
29 \( 1 - 9.02T + 29T^{2} \)
31 \( 1 - 6.57T + 31T^{2} \)
37 \( 1 - 0.619T + 37T^{2} \)
41 \( 1 - 8.41T + 41T^{2} \)
43 \( 1 + 8.50T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 9.83T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.358871707737911036363209437755, −8.050110558562337889497341816640, −7.43764424417176688368030019544, −6.60306885887056381332976161329, −5.49737446793876397328379580692, −4.59550626972253482097166401877, −3.64161474167353592011634231594, −2.98711107324965190837821119660, −2.43002034324168670057367512959, −0.978912672935551598642057857915, 0.978912672935551598642057857915, 2.43002034324168670057367512959, 2.98711107324965190837821119660, 3.64161474167353592011634231594, 4.59550626972253482097166401877, 5.49737446793876397328379580692, 6.60306885887056381332976161329, 7.43764424417176688368030019544, 8.050110558562337889497341816640, 8.358871707737911036363209437755

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