Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.999 - 0.0387i$
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 + (−3.43 + 3.43i)13-s + 1.00·14-s − 1.00·16-s + (0.906 − 0.906i)17-s − 2.04i·19-s + (0.414 + 0.414i)22-s + (−0.257 − 0.257i)23-s − 4.86i·26-s + (−0.707 + 0.707i)28-s + 1.75·29-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 + (−0.953 + 0.953i)13-s + 0.267·14-s − 0.250·16-s + (0.219 − 0.219i)17-s − 0.470i·19-s + (0.0883 + 0.0883i)22-s + (−0.0537 − 0.0537i)23-s − 0.953i·26-s + (−0.133 + 0.133i)28-s + 0.325·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.0387i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.999 - 0.0387i)$
$L(1)$  $\approx$  $1.121564243$
$L(\frac12)$  $\approx$  $1.121564243$
$L(\frac{3}{2})$   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 \( 1 + (3.43 - 3.43i)T - 13iT^{2} \)
17 \( 1 + (-0.906 + 0.906i)T - 17iT^{2} \)
19 \( 1 + 2.04iT - 19T^{2} \)
23 \( 1 + (0.257 + 0.257i)T + 23iT^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 - 1.47T + 31T^{2} \)
37 \( 1 + (-4.04 - 4.04i)T + 37iT^{2} \)
41 \( 1 + 3.84iT - 41T^{2} \)
43 \( 1 + (5.10 - 5.10i)T - 43iT^{2} \)
47 \( 1 + (-5.49 + 5.49i)T - 47iT^{2} \)
53 \( 1 + (5.02 + 5.02i)T + 53iT^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 3.78T + 61T^{2} \)
67 \( 1 + (-6.74 - 6.74i)T + 67iT^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (5.97 - 5.97i)T - 73iT^{2} \)
79 \( 1 + 0.944iT - 79T^{2} \)
83 \( 1 + (-7.82 - 7.82i)T + 83iT^{2} \)
89 \( 1 - 0.0705T + 89T^{2} \)
97 \( 1 + (-0.746 - 0.746i)T + 97iT^{2} \)
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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

−8.607939050287288016438270917200, −7.988552528446080432047316521128, −6.98016261284486714441154568674, −6.79460264218925820319301395069, −5.73799741330026210888298053556, −4.90448563380761616764360616930, −4.15777589451826408215959423702, −2.96360891159280893940080744495, −1.94637393172584072472016517687, −0.59527336924763467924883553242, 0.77592535775976521421220301103, 2.10758038220133630552309515733, 2.89598299012985863014463256416, 3.77188498422400026478529536334, 4.78961337563564726228340486511, 5.62125663045490802222096777189, 6.47995063762519355083123325795, 7.44822949888239034101951556581, 7.941164031535419761884902130393, 8.741182750857460192288702023368

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