Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.924 - 0.381i$
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 + (−1.41 − 2.23i)7-s + i·8-s + 1.41i·11-s + 5.39i·13-s + (−2.23 + 1.41i)14-s + 16-s + 2.23·17-s − 1.30i·19-s + 1.41·22-s + i·23-s + 5.39·26-s + (1.41 + 2.23i)28-s − 9.24i·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.534 − 0.845i)7-s + 0.353i·8-s + 0.426i·11-s + 1.49i·13-s + (−0.597 + 0.377i)14-s + 0.250·16-s + 0.542·17-s − 0.300i·19-s + 0.301·22-s + 0.208i·23-s + 1.05·26-s + (0.267 + 0.422i)28-s − 1.71i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.924 - 0.381i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.924 - 0.381i)$
$L(1)$  $\approx$  $0.4600411194$
$L(\frac12)$  $\approx$  $0.4600411194$
$L(\frac{3}{2})$   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 \( 1 + (1.41 + 2.23i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 5.39iT - 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 + 1.30iT - 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 9.24iT - 29T^{2} \)
31 \( 1 + 8.56iT - 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + 4.08T + 41T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 + 7.63T + 47T^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 5.39iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 4.34iT - 71T^{2} \)
73 \( 1 - 5.01iT - 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 + 8.01T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−8.258204178486945456199811865530, −7.59052952530119633310623100053, −6.72540972807836003794013006179, −6.10043236153046114507167870472, −4.87620981011954607821326952650, −4.20438733701240663663257056850, −3.55898523683382855185636165313, −2.44049991192942436101943088301, −1.49848343958977023737306115885, −0.14658023266366935895210146703, 1.37284302535930975881337069761, 3.06018448128250470943983965555, 3.30313547848045909598391940188, 4.78939280708426255813509582845, 5.44705226380228858756615347241, 6.00782131674365299515191412426, 6.81852873576031437608627521825, 7.60507103053431067533028675859, 8.507506026427530618225728938046, 8.729569389265981161401867014222

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