Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.836 + 0.548i$
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 + (−1.52 + 2.16i)7-s − 0.999i·8-s + (4.29 + 2.48i)11-s − 5.49i·13-s + (−0.243 + 2.63i)14-s + (−0.5 − 0.866i)16-s + (1.53 − 2.66i)17-s + (−2.68 + 1.55i)19-s + 4.96·22-s + (5.34 − 3.08i)23-s + (−2.74 − 4.75i)26-s + (1.10 + 2.40i)28-s + 6.67i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.577 + 0.816i)7-s − 0.353i·8-s + (1.29 + 0.748i)11-s − 1.52i·13-s + (−0.0649 + 0.704i)14-s + (−0.125 − 0.216i)16-s + (0.372 − 0.645i)17-s + (−0.616 + 0.355i)19-s + 1.05·22-s + (1.11 − 0.643i)23-s + (−0.538 − 0.933i)26-s + (0.209 + 0.454i)28-s + 1.24i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.836 + 0.548i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.836 + 0.548i)$
$L(1)$  $\approx$  $2.753230718$
$L(\frac12)$  $\approx$  $2.753230718$
$L(\frac{3}{2})$   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 \( 1 + (1.52 - 2.16i)T \)
good11 \( 1 + (-4.29 - 2.48i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.49iT - 13T^{2} \)
17 \( 1 + (-1.53 + 2.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.68 - 1.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.34 + 3.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.67iT - 29T^{2} \)
31 \( 1 + (1.01 + 0.586i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.35 - 9.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.39T + 41T^{2} \)
43 \( 1 - 8.81T + 43T^{2} \)
47 \( 1 + (2.07 + 3.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.85 + 2.22i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.00 + 5.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.05 + 5.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.97 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.973iT - 71T^{2} \)
73 \( 1 + (-14.4 - 8.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.12 - 3.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + (-7.38 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.41iT - 97T^{2} \)
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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

−8.704871807769288342058160568648, −7.896287300958883658820126893238, −6.70328140909910746724832699928, −6.49703895674716651040075472823, −5.32233608451652315604852904373, −4.92851587808136695618686445168, −3.69397610800376530992136682730, −3.09439109337920851597694762530, −2.15525260893259018286574749203, −0.899035028728805771103260176092, 0.992502367640058237224940705816, 2.25687149700530949490796812156, 3.61325880019880887447345227395, 3.92141239880826394687340589360, 4.74794767050687500814608216828, 5.98767121872612823290184144070, 6.41039441455574564817226545592, 7.07437013198325687198550014523, 7.78081004069247478304951000793, 8.947153250916236371075464917239

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