Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.492 - 0.870i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−2.23 − 1.41i)7-s + 8-s + 1.41i·11-s − 5.39·13-s + (−2.23 − 1.41i)14-s + 16-s − 2.23i·17-s − 1.30i·19-s + 1.41i·22-s + 23-s − 5.39·26-s + (−2.23 − 1.41i)28-s + 9.24i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.845 − 0.534i)7-s + 0.353·8-s + 0.426i·11-s − 1.49·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s − 0.300i·19-s + 0.301i·22-s + 0.208·23-s − 1.05·26-s + (−0.422 − 0.267i)28-s + 1.71i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.492 - 0.870i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.492 - 0.870i)$
$L(1)$  $\approx$  $1.104797297$
$L(\frac12)$  $\approx$  $1.104797297$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.23 + 1.41i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 5.39T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + 1.30iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - 9.24iT - 29T^{2} \)
31 \( 1 - 8.56iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 - 4.08T + 41T^{2} \)
43 \( 1 - 6.41iT - 43T^{2} \)
47 \( 1 - 7.63iT - 47T^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 5.39iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 4.34iT - 71T^{2} \)
73 \( 1 + 5.01T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 8.01iT - 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−9.166611544056358035720715383692, −7.934531822071082413479009251490, −7.08406993035804116759588512828, −6.89625873310842114775374219359, −5.85755055993511969299193099833, −4.85309088737823201230750857112, −4.51497649071041394476138308521, −3.19299183418908177482572634952, −2.80454516223016457762985062587, −1.40505562246421168005337828059, 0.24824282207563417968294495053, 2.10180467581432643270710542935, 2.73967783969234922599297110963, 3.73560250205082956434627126805, 4.48512242824161871768873289796, 5.53284874899074038193057976610, 5.98199176503091750432318755332, 6.79797531642429498734267134810, 7.60358376042590070124177858308, 8.294791306159265906749238119579

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