Properties

Label 2-4598-1.1-c1-0-113
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.76·3-s + 4-s + 1.76·5-s + 2.76·6-s + 3.62·7-s − 8-s + 4.62·9-s − 1.76·10-s − 2.76·12-s − 2.62·13-s − 3.62·14-s − 4.86·15-s + 16-s + 4.62·17-s − 4.62·18-s + 19-s + 1.76·20-s − 10.0·21-s + 3.62·23-s + 2.76·24-s − 1.89·25-s + 2.62·26-s − 4.49·27-s + 3.62·28-s − 5.38·29-s + 4.86·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.59·3-s + 0.5·4-s + 0.787·5-s + 1.12·6-s + 1.37·7-s − 0.353·8-s + 1.54·9-s − 0.557·10-s − 0.797·12-s − 0.728·13-s − 0.969·14-s − 1.25·15-s + 0.250·16-s + 1.12·17-s − 1.09·18-s + 0.229·19-s + 0.393·20-s − 2.18·21-s + 0.756·23-s + 0.563·24-s − 0.379·25-s + 0.515·26-s − 0.864·27-s + 0.685·28-s − 1.00·29-s + 0.888·30-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: \(1\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
11 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 2.76T + 3T^{2} \)
5 \( 1 - 1.76T + 5T^{2} \)
7 \( 1 - 3.62T + 7T^{2} \)
13 \( 1 + 2.62T + 13T^{2} \)
17 \( 1 - 4.62T + 17T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + 5.38T + 29T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 + 2.61T + 37T^{2} \)
41 \( 1 + 5.01T + 41T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 + 7.42T + 47T^{2} \)
53 \( 1 + 5.11T + 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 + 4.59T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 + 0.864T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 - 6.38T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 1.10T + 97T^{2} \)
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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

−7.72770113325717970490614137500, −7.35642459149097502108349603353, −6.40168905986566553316304154583, −5.75685722716459195213294401162, −5.11813565057972743002542555834, −4.70320609011292927978623624515, −3.24107879349488428431191971090, −1.80637122715310310147344900436, −1.36839823670038574094640681421, 0, 1.36839823670038574094640681421, 1.80637122715310310147344900436, 3.24107879349488428431191971090, 4.70320609011292927978623624515, 5.11813565057972743002542555834, 5.75685722716459195213294401162, 6.40168905986566553316304154583, 7.35642459149097502108349603353, 7.72770113325717970490614137500

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