Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
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-s + 0.447·3-s + 4-s − 5-s + 0.447·6-s + 1.86·7-s + 8-s − 2.79·9-s − 10-s − 4.77·11-s + 0.447·12-s − 13-s + 1.86·14-s − 0.447·15-s + 16-s + 0.778·17-s − 2.79·18-s + 1.72·19-s − 20-s + 0.834·21-s − 4.77·22-s + 7.35·23-s + 0.447·24-s + 25-s − 26-s − 2.59·27-s + 1.86·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.258·3-s + 0.5·4-s − 0.447·5-s + 0.182·6-s + 0.704·7-s + 0.353·8-s − 0.933·9-s − 0.316·10-s − 1.43·11-s + 0.129·12-s − 0.277·13-s + 0.498·14-s − 0.115·15-s + 0.250·16-s + 0.188·17-s − 0.659·18-s + 0.394·19-s − 0.223·20-s + 0.182·21-s − 1.01·22-s + 1.53·23-s + 0.0913·24-s + 0.200·25-s − 0.196·26-s − 0.499·27-s + 0.352·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.879555832$
$L(\frac12)$  $\approx$  $2.879555832$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
31 \( 1 + T \)
good3 \( 1 - 0.447T + 3T^{2} \)
7 \( 1 - 1.86T + 7T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
17 \( 1 - 0.778T + 17T^{2} \)
19 \( 1 - 1.72T + 19T^{2} \)
23 \( 1 - 7.35T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 - 9.99T + 41T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 0.881T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 3.07T + 73T^{2} \)
79 \( 1 + 6.67T + 79T^{2} \)
83 \( 1 - 9.02T + 83T^{2} \)
89 \( 1 - 1.00T + 89T^{2} \)
97 \( 1 + 0.347T + 97T^{2} \)
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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

−8.318477914587690933476880408649, −7.61946538554596665882339629274, −7.18735764263874096947848312628, −5.98496444403688986664534046403, −5.31648814249654893504315857916, −4.81067055822541395898596609113, −3.86044413310342055539829176018, −2.79817729689008600843588397467, −2.47929094525654821875115895068, −0.854785005844326845476791014616, 0.854785005844326845476791014616, 2.47929094525654821875115895068, 2.79817729689008600843588397467, 3.86044413310342055539829176018, 4.81067055822541395898596609113, 5.31648814249654893504315857916, 5.98496444403688986664534046403, 7.18735764263874096947848312628, 7.61946538554596665882339629274, 8.318477914587690933476880408649

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