Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.894 + 0.447i$
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 i·7-s + i·8-s − 4·11-s + 3i·13-s − 14-s + 16-s + 7i·17-s + 6·19-s + 4i·22-s − 9i·23-s + 3·26-s + i·28-s + 3·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 1.20·11-s + 0.832i·13-s − 0.267·14-s + 0.250·16-s + 1.69i·17-s + 1.37·19-s + 0.852i·22-s − 1.87i·23-s + 0.588·26-s + 0.188i·28-s + 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.894 + 0.447i)$
$L(1)$  $\approx$  $0.9334525052$
$L(\frac12)$  $\approx$  $0.9334525052$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 + 13iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - iT - 53T^{2} \)
59 \( 1 + 11T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.520889850402485317293226546352, −7.68838988493980913782514098621, −6.98139296856741614280793876088, −5.93877688352697497386510716128, −5.21222496618157481439847291050, −4.28051341713366658861968841436, −3.62037561762735587466915906076, −2.54719009480443525736089113299, −1.71255675964978544711776484781, −0.31349598183089335991074913843, 1.15903430291555601009020863450, 2.80854997288585648059657907601, 3.27121004810812307062897605064, 4.73836620825753519083624131748, 5.31930590483731274998713153941, 5.74843584378754269945124412371, 6.91511882506102448207047928237, 7.64528950345514388191840347560, 7.924726274951671188813213117863, 8.971814274350725086893985766304

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