# Properties

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

# Related objects

## 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}$$
show less
\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