Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.161 - 0.986i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + 0.585i·11-s + (0.0249 + 0.0249i)13-s + 1.00·14-s − 1.00·16-s + (1.92 + 1.92i)17-s − 4.87i·19-s + (0.414 − 0.414i)22-s + (−5.15 + 5.15i)23-s − 0.0352i·26-s + (−0.707 − 0.707i)28-s − 4.58·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + 0.176i·11-s + (0.00691 + 0.00691i)13-s + 0.267·14-s − 0.250·16-s + (0.466 + 0.466i)17-s − 1.11i·19-s + (0.0883 − 0.0883i)22-s + (−1.07 + 1.07i)23-s − 0.00691i·26-s + (−0.133 − 0.133i)28-s − 0.850·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.161 - 0.986i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.161 - 0.986i)$
$L(1)$  $\approx$  $0.6446856571$
$L(\frac12)$  $\approx$  $0.6446856571$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 0.585iT - 11T^{2} \)
13 \( 1 + (-0.0249 - 0.0249i)T + 13iT^{2} \)
17 \( 1 + (-1.92 - 1.92i)T + 17iT^{2} \)
19 \( 1 + 4.87iT - 19T^{2} \)
23 \( 1 + (5.15 - 5.15i)T - 23iT^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 - 9.83T + 31T^{2} \)
37 \( 1 + (2.87 - 2.87i)T - 37iT^{2} \)
41 \( 1 - 0.979iT - 41T^{2} \)
43 \( 1 + (-4.27 - 4.27i)T + 43iT^{2} \)
47 \( 1 + (6.32 + 6.32i)T + 47iT^{2} \)
53 \( 1 + (1.56 - 1.56i)T - 53iT^{2} \)
59 \( 1 + 0.670T + 59T^{2} \)
61 \( 1 - 2.35T + 61T^{2} \)
67 \( 1 + (5.08 - 5.08i)T - 67iT^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (2.51 + 2.51i)T + 73iT^{2} \)
79 \( 1 - 8.71iT - 79T^{2} \)
83 \( 1 + (2.99 - 2.99i)T - 83iT^{2} \)
89 \( 1 + 9.72T + 89T^{2} \)
97 \( 1 + (4.74 - 4.74i)T - 97iT^{2} \)
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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

−8.940008219184125110325694160187, −8.227264947012085456281021409136, −7.56909865034873018096077149346, −6.71403116899576171679195506691, −5.93166836681660082157209108015, −4.99095786809696664814019809481, −4.04769307320253296421029804457, −3.18216560597100365280865005545, −2.30114010135297293927146031234, −1.22682663815592203146025182891, 0.25124197408795609729630459576, 1.54160475716783851863535452208, 2.72790199250712783389672063311, 3.81595743108613080079582295115, 4.64781373861137568291922661972, 5.69433953484798156510286866639, 6.22214853884821933471753233298, 7.03771085726985511381954565466, 7.83960886127273342503509187591, 8.321299971378780623128084092083

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