Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.309 - 0.950i$
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 + (−2.12 + 1.58i)7-s i·8-s + 1.41i·11-s − 3.16i·13-s + (−1.58 − 2.12i)14-s + 16-s + 4.47·17-s − 1.41·22-s − 6i·23-s + 3.16·26-s + (2.12 − 1.58i)28-s + 2.82i·29-s + i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.801 + 0.597i)7-s − 0.353i·8-s + 0.426i·11-s − 0.877i·13-s + (−0.422 − 0.566i)14-s + 0.250·16-s + 1.08·17-s − 0.301·22-s − 1.25i·23-s + 0.620·26-s + (0.400 − 0.298i)28-s + 0.525i·29-s + 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.309 - 0.950i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.309 - 0.950i)$
$L(1)$  $\approx$  $1.471354910$
$L(\frac12)$  $\approx$  $1.471354910$
$L(\frac{3}{2})$   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 \( 1 + (2.12 - 1.58i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 3.16iT - 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 9.48T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 - 4.47T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 - 6.32iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−8.660132650359010560573903910052, −8.135565099030279020115821542908, −7.23339823549306703299757511009, −6.62994323415416782856492017483, −5.73779314262574789108775621242, −5.30540466022121611294662912513, −4.22822934998891776904029256027, −3.28885593833010005628792800811, −2.43559521041802132733773474054, −0.799780946583711719781980224577, 0.67798866511102393119037935785, 1.79810788007788625237553595835, 3.01818952441766136096994608524, 3.65387939743807060257785817763, 4.40454637481560450660674396891, 5.47922098466323583839562754438, 6.18897252039217780966329054809, 7.12070260282011359611503176203, 7.79189375727999777484225087868, 8.683707661707473615537797972849

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