Properties

Degree 2
Conductor $ 7 \cdot 859 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.31·2-s + 3.03·3-s + 3.37·4-s + 1.64·5-s − 7.02·6-s + 7-s − 3.19·8-s + 6.18·9-s − 3.81·10-s + 1.18·11-s + 10.2·12-s + 5.12·13-s − 2.31·14-s + 4.99·15-s + 0.658·16-s − 6.39·17-s − 14.3·18-s + 6.08·19-s + 5.56·20-s + 3.03·21-s − 2.74·22-s + 5.34·23-s − 9.69·24-s − 2.28·25-s − 11.8·26-s + 9.66·27-s + 3.37·28-s + ⋯
L(s)  = 1  − 1.63·2-s + 1.75·3-s + 1.68·4-s + 0.736·5-s − 2.86·6-s + 0.377·7-s − 1.13·8-s + 2.06·9-s − 1.20·10-s + 0.357·11-s + 2.95·12-s + 1.42·13-s − 0.619·14-s + 1.28·15-s + 0.164·16-s − 1.55·17-s − 3.38·18-s + 1.39·19-s + 1.24·20-s + 0.661·21-s − 0.585·22-s + 1.11·23-s − 1.97·24-s − 0.457·25-s − 2.33·26-s + 1.85·27-s + 0.638·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6013\)    =    \(7 \cdot 859\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6013} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6013,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.562803991\)
\(L(\frac12)\)  \(\approx\)  \(2.562803991\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;859\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;859\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - T \)
859 \( 1 + T \)
good2 \( 1 + 2.31T + 2T^{2} \)
3 \( 1 - 3.03T + 3T^{2} \)
5 \( 1 - 1.64T + 5T^{2} \)
11 \( 1 - 1.18T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 6.39T + 17T^{2} \)
19 \( 1 - 6.08T + 19T^{2} \)
23 \( 1 - 5.34T + 23T^{2} \)
29 \( 1 - 7.62T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 7.75T + 37T^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 - 5.89T + 53T^{2} \)
59 \( 1 - 6.73T + 59T^{2} \)
61 \( 1 - 4.52T + 61T^{2} \)
67 \( 1 + 0.200T + 67T^{2} \)
71 \( 1 - 7.70T + 71T^{2} \)
73 \( 1 + 2.54T + 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 - 4.01T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 6.33T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.333043292719113238801500910703, −7.74631407647556860572296305813, −6.90825314829507570983543307409, −6.52718171698620275654152318596, −5.26109902865442206391786180055, −4.14006227639445212830399607838, −3.28316827535429008284210463319, −2.43143049754814520699333461309, −1.72067063145866473446762377544, −1.10236590034814916569356880610, 1.10236590034814916569356880610, 1.72067063145866473446762377544, 2.43143049754814520699333461309, 3.28316827535429008284210463319, 4.14006227639445212830399607838, 5.26109902865442206391786180055, 6.52718171698620275654152318596, 6.90825314829507570983543307409, 7.74631407647556860572296305813, 8.333043292719113238801500910703

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