Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.933 - 0.359i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (2.56 + 0.648i)7-s i·8-s − 1.29i·11-s + 3.13i·13-s + (−0.648 + 2.56i)14-s + 16-s + 5.53·17-s − 7.37i·19-s + 1.29·22-s − 1.83i·23-s − 3.13·26-s + (−2.56 − 0.648i)28-s + 1.83i·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.969 + 0.245i)7-s − 0.353i·8-s − 0.390i·11-s + 0.868i·13-s + (−0.173 + 0.685i)14-s + 0.250·16-s + 1.34·17-s − 1.69i·19-s + 0.276·22-s − 0.382i·23-s − 0.613·26-s + (−0.484 − 0.122i)28-s + 0.340i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.933 - 0.359i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.933 - 0.359i)$
$L(1)$  $\approx$  $2.014125658$
$L(\frac12)$  $\approx$  $2.014125658$
$L(\frac{3}{2})$   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 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.56 - 0.648i)T \)
good11 \( 1 + 1.29iT - 11T^{2} \)
13 \( 1 - 3.13iT - 13T^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 + 7.37iT - 19T^{2} \)
23 \( 1 + 1.83iT - 23T^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 3.13T + 41T^{2} \)
43 \( 1 + 3.53T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 4.42iT - 53T^{2} \)
59 \( 1 + 7.18T + 59T^{2} \)
61 \( 1 - 4.88iT - 61T^{2} \)
67 \( 1 - 9.79T + 67T^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 - 3.40iT - 73T^{2} \)
79 \( 1 - 9.01T + 79T^{2} \)
83 \( 1 + 6.26T + 83T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + 8.09iT - 97T^{2} \)
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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

−8.537310900084858749769378396334, −7.982795710818231639725453327525, −7.29664228162517516694778965913, −6.46459527812425708798317724480, −5.72968533825558750962528221471, −4.88698347486874672314477674820, −4.35287134287515595053302353765, −3.19193278162007376066262115585, −2.06238771837244040127523595561, −0.76447652619687584182515653060, 1.08898612107020500047483447233, 1.80928622493081156714518889660, 3.09557148168930825679279819238, 3.74663061951671635666622452976, 4.79592734900566179206976965521, 5.36681924551424969954378347512, 6.22018495807297174051101997979, 7.47386370069462294678182582764, 7.989821889050953579009372320702, 8.479081630445255267292416585877

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