Properties

Label 2-1617-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.36·2-s + 3-s + 3.57·4-s + 3.93·5-s − 2.36·6-s − 3.72·8-s + 9-s − 9.29·10-s + 11-s + 3.57·12-s + 3.93·13-s + 3.93·15-s + 1.63·16-s − 4.72·17-s − 2.36·18-s − 4.78·19-s + 14.0·20-s − 2.36·22-s + 2.72·23-s − 3.72·24-s + 10.5·25-s − 9.29·26-s + 27-s + 7.93·29-s − 9.29·30-s − 1.15·31-s + 3.57·32-s + ⋯
L(s)  = 1  − 1.66·2-s + 0.577·3-s + 1.78·4-s + 1.76·5-s − 0.964·6-s − 1.31·8-s + 0.333·9-s − 2.94·10-s + 0.301·11-s + 1.03·12-s + 1.09·13-s + 1.01·15-s + 0.409·16-s − 1.14·17-s − 0.556·18-s − 1.09·19-s + 3.14·20-s − 0.503·22-s + 0.567·23-s − 0.759·24-s + 2.10·25-s − 1.82·26-s + 0.192·27-s + 1.47·29-s − 1.69·30-s − 0.207·31-s + 0.632·32-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: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.423652853\)
\(L(\frac12)\) \(\approx\) \(1.423652853\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.36T + 2T^{2} \)
5 \( 1 - 3.93T + 5T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 + 4.72T + 17T^{2} \)
19 \( 1 + 4.78T + 19T^{2} \)
23 \( 1 - 2.72T + 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 1.15T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 + 0.430T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 - 8.78T + 47T^{2} \)
53 \( 1 + 3.15T + 53T^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 3.21T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 5.44T + 79T^{2} \)
83 \( 1 - 2.84T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−9.254148076137820904158183610331, −8.726587465341917362330223519376, −8.339291611025795272185848833627, −6.90337640803366801615183701374, −6.60536666030595618066818356139, −5.68015039869709883815369532149, −4.31283768562594336707138210583, −2.72056556367129788393911038859, −2.01360373016954531569139020812, −1.12924938233690040378552685227, 1.12924938233690040378552685227, 2.01360373016954531569139020812, 2.72056556367129788393911038859, 4.31283768562594336707138210583, 5.68015039869709883815369532149, 6.60536666030595618066818356139, 6.90337640803366801615183701374, 8.339291611025795272185848833627, 8.726587465341917362330223519376, 9.254148076137820904158183610331

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