Properties

Label 2-4598-1.1-c1-0-88
Degree $2$
Conductor $4598$
Sign $1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
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-s + 2.66·3-s + 4-s − 4.12·5-s + 2.66·6-s + 4.21·7-s + 8-s + 4.12·9-s − 4.12·10-s + 2.66·12-s + 2.21·13-s + 4.21·14-s − 11.0·15-s + 16-s + 3.45·17-s + 4.12·18-s + 19-s − 4.12·20-s + 11.2·21-s − 5.45·23-s + 2.66·24-s + 12.0·25-s + 2.21·26-s + 3.00·27-s + 4.21·28-s − 5.57·29-s − 11.0·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.54·3-s + 0.5·4-s − 1.84·5-s + 1.08·6-s + 1.59·7-s + 0.353·8-s + 1.37·9-s − 1.30·10-s + 0.770·12-s + 0.614·13-s + 1.12·14-s − 2.84·15-s + 0.250·16-s + 0.837·17-s + 0.972·18-s + 0.229·19-s − 0.922·20-s + 2.45·21-s − 1.13·23-s + 0.544·24-s + 2.40·25-s + 0.434·26-s + 0.577·27-s + 0.796·28-s − 1.03·29-s − 2.00·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4598\)    =    \(2 \cdot 11^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(36.7152\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4598} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.068685760\)
\(L(\frac12)\) \(\approx\) \(5.068685760\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 2.66T + 3T^{2} \)
5 \( 1 + 4.12T + 5T^{2} \)
7 \( 1 - 4.21T + 7T^{2} \)
13 \( 1 - 2.21T + 13T^{2} \)
17 \( 1 - 3.45T + 17T^{2} \)
23 \( 1 + 5.45T + 23T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
31 \( 1 - 7.00T + 31T^{2} \)
37 \( 1 + 2.90T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 - 7.58T + 47T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 - 4.79T + 59T^{2} \)
61 \( 1 + 8.90T + 61T^{2} \)
67 \( 1 - 1.30T + 67T^{2} \)
71 \( 1 + 6.80T + 71T^{2} \)
73 \( 1 - 1.45T + 73T^{2} \)
79 \( 1 - 9.15T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 8.24T + 89T^{2} \)
97 \( 1 - 2.18T + 97T^{2} \)
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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

−8.125122928317021842148559548627, −7.65080795025105146044204539790, −7.44922966589162054389085277794, −6.11913326865801565796274722135, −4.99235116511441522350164379501, −4.29817108437343017689720436166, −3.80101491747789085793622769874, −3.17069275329707419667172218240, −2.18174764107316283066687412850, −1.14323486462799679468290454975, 1.14323486462799679468290454975, 2.18174764107316283066687412850, 3.17069275329707419667172218240, 3.80101491747789085793622769874, 4.29817108437343017689720436166, 4.99235116511441522350164379501, 6.11913326865801565796274722135, 7.44922966589162054389085277794, 7.65080795025105146044204539790, 8.125122928317021842148559548627

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