Properties

Label 2-1617-1.1-c1-0-38
Degree $2$
Conductor $1617$
Sign $-1$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  − 2.41·2-s + 3-s + 3.82·4-s − 2·5-s − 2.41·6-s − 4.41·8-s + 9-s + 4.82·10-s + 11-s + 3.82·12-s − 0.828·13-s − 2·15-s + 2.99·16-s − 4.41·17-s − 2.41·18-s + 7.24·19-s − 7.65·20-s − 2.41·22-s − 7·23-s − 4.41·24-s − 25-s + 1.99·26-s + 27-s + 3.24·29-s + 4.82·30-s + 5.65·31-s + 1.58·32-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.91·4-s − 0.894·5-s − 0.985·6-s − 1.56·8-s + 0.333·9-s + 1.52·10-s + 0.301·11-s + 1.10·12-s − 0.229·13-s − 0.516·15-s + 0.749·16-s − 1.07·17-s − 0.569·18-s + 1.66·19-s − 1.71·20-s − 0.514·22-s − 1.45·23-s − 0.901·24-s − 0.200·25-s + 0.392·26-s + 0.192·27-s + 0.602·29-s + 0.881·30-s + 1.01·31-s + 0.280·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 - 7.24T + 19T^{2} \)
23 \( 1 + 7T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + 1.17T + 41T^{2} \)
43 \( 1 + 2.75T + 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 + 7.17T + 53T^{2} \)
59 \( 1 + 8.65T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 3.17T + 67T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 - 0.343T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.925244273161343507784713384625, −8.274631162935441020542379527799, −7.66628361659559764926271557384, −7.08900222400342542256647763104, −6.18925706956845655307866109056, −4.69304038187189757170150256612, −3.63038134202443877586541385678, −2.55269324795494843239122022897, −1.41942960474794095224470410930, 0, 1.41942960474794095224470410930, 2.55269324795494843239122022897, 3.63038134202443877586541385678, 4.69304038187189757170150256612, 6.18925706956845655307866109056, 7.08900222400342542256647763104, 7.66628361659559764926271557384, 8.274631162935441020542379527799, 8.925244273161343507784713384625

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