Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.972 - 0.231i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (2.63 + 0.189i)7-s + 0.999i·8-s + (4.67 − 2.69i)11-s − 2.51i·13-s + (2.19 + 1.48i)14-s + (−0.5 + 0.866i)16-s + (2.24 + 3.89i)17-s + (−2.48 − 1.43i)19-s + 5.39·22-s + (0.232 + 0.133i)23-s + (1.25 − 2.18i)26-s + (1.15 + 2.38i)28-s − 8.89i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.997 + 0.0716i)7-s + 0.353i·8-s + (1.40 − 0.813i)11-s − 0.698i·13-s + (0.585 + 0.396i)14-s + (−0.125 + 0.216i)16-s + (0.545 + 0.945i)17-s + (−0.568 − 0.328i)19-s + 1.15·22-s + (0.0483 + 0.0279i)23-s + (0.246 − 0.427i)26-s + (0.218 + 0.449i)28-s − 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.972 - 0.231i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.972 - 0.231i)$
$L(1)$  $\approx$  $3.410197034$
$L(\frac12)$  $\approx$  $3.410197034$
$L(\frac{3}{2})$   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 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.63 - 0.189i)T \)
good11 \( 1 + (-4.67 + 2.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.51iT - 13T^{2} \)
17 \( 1 + (-2.24 - 3.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.48 + 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.232 - 0.133i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.25 - 5.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.760T + 41T^{2} \)
43 \( 1 - 5.86T + 43T^{2} \)
47 \( 1 + (-3.99 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.27 - 4.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.33 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (10.0 - 5.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.29 - 7.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + (-3.98 + 6.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.16iT - 97T^{2} \)
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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

−8.344440947451061409186346705978, −8.145531878105953357175562832905, −7.15877360326066926881171441547, −6.16133314356926365306737001404, −5.87878381006405849963267041398, −4.78802794622592634822674047456, −4.10475274433088590661407746465, −3.33402712191915878305625853260, −2.15986903264985027112467001917, −1.01768218163845986120464682904, 1.25077375353777933346491739195, 1.90723850324376831958447251358, 3.09837895885299246185249207642, 4.15826323398127330277547961002, 4.59934400997275914258241498441, 5.41978160574850967261949034324, 6.40777888761853883387238004928, 7.07921705520863475918737097274, 7.73213356158928520438601144323, 8.948313272733850174634225295022

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