Properties

Label 2-7616-1.1-c1-0-105
Degree $2$
Conductor $7616$
Sign $1$
Analytic cond. $60.8140$
Root an. cond. $7.79833$
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  + 1.90·3-s + 2.90·5-s − 7-s + 0.618·9-s + 2.79·11-s + 1.95·13-s + 5.52·15-s + 17-s − 4.42·19-s − 1.90·21-s + 1.72·23-s + 3.42·25-s − 4.53·27-s − 8.15·29-s + 9.64·31-s + 5.31·33-s − 2.90·35-s + 2.61·37-s + 3.71·39-s − 1.43·41-s + 8.99·43-s + 1.79·45-s + 2.65·47-s + 49-s + 1.90·51-s + 4.50·53-s + 8.10·55-s + ⋯
L(s)  = 1  + 1.09·3-s + 1.29·5-s − 0.377·7-s + 0.206·9-s + 0.842·11-s + 0.541·13-s + 1.42·15-s + 0.242·17-s − 1.01·19-s − 0.415·21-s + 0.360·23-s + 0.684·25-s − 0.871·27-s − 1.51·29-s + 1.73·31-s + 0.925·33-s − 0.490·35-s + 0.429·37-s + 0.594·39-s − 0.224·41-s + 1.37·43-s + 0.267·45-s + 0.387·47-s + 0.142·49-s + 0.266·51-s + 0.618·53-s + 1.09·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7616\)    =    \(2^{6} \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(60.8140\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7616,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.163583163\)
\(L(\frac12)\) \(\approx\) \(4.163583163\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
17 \( 1 - T \)
good3 \( 1 - 1.90T + 3T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
11 \( 1 - 2.79T + 11T^{2} \)
13 \( 1 - 1.95T + 13T^{2} \)
19 \( 1 + 4.42T + 19T^{2} \)
23 \( 1 - 1.72T + 23T^{2} \)
29 \( 1 + 8.15T + 29T^{2} \)
31 \( 1 - 9.64T + 31T^{2} \)
37 \( 1 - 2.61T + 37T^{2} \)
41 \( 1 + 1.43T + 41T^{2} \)
43 \( 1 - 8.99T + 43T^{2} \)
47 \( 1 - 2.65T + 47T^{2} \)
53 \( 1 - 4.50T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 8.64T + 71T^{2} \)
73 \( 1 - 2.37T + 73T^{2} \)
79 \( 1 - 8.59T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−8.052339268933643340096031021568, −7.18617137714968631093070196980, −6.30742426914204626502146480120, −6.04408708699242834405831665720, −5.13195312169683749382178793611, −4.05129539922926690655285107763, −3.50815935918370658877403873936, −2.49840489940042788632842785416, −2.07153974051976643135584399764, −1.00029120283349766712854490917, 1.00029120283349766712854490917, 2.07153974051976643135584399764, 2.49840489940042788632842785416, 3.50815935918370658877403873936, 4.05129539922926690655285107763, 5.13195312169683749382178793611, 6.04408708699242834405831665720, 6.30742426914204626502146480120, 7.18617137714968631093070196980, 8.052339268933643340096031021568

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