Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.100 + 0.994i$
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 + (−2.62 + 0.358i)7-s − 0.999i·8-s + (−2.59 + 1.5i)11-s + 2.44i·13-s + (2.44 + i)14-s + (−0.5 + 0.866i)16-s + (2.95 + 5.12i)17-s + (−5.12 − 2.95i)19-s + 3·22-s + (−3.67 − 2.12i)23-s + (1.22 − 2.12i)26-s + (−1.62 − 2.09i)28-s − 7.24i·29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.990 + 0.135i)7-s − 0.353i·8-s + (−0.783 + 0.452i)11-s + 0.679i·13-s + (0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.717 + 1.24i)17-s + (−1.17 − 0.678i)19-s + 0.639·22-s + (−0.766 − 0.442i)23-s + (0.240 − 0.416i)26-s + (−0.306 − 0.395i)28-s − 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.100 + 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.100 + 0.994i)$
$L(1)$  $\approx$  $0.5787764919$
$L(\frac12)$  $\approx$  $0.5787764919$
$L(\frac{3}{2})$   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 \( 1 + (2.62 - 0.358i)T \)
good11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-2.95 - 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.121 - 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 + (-2.95 + 5.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.27 + 3.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.03 - 6.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (-1.24 + 0.717i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−8.684092342216333875566000649297, −7.82283273346788174181445636803, −7.12376792539545159599590697112, −6.29918796144538775317491725389, −5.67859243494536862018872342024, −4.36059444641613049764801519917, −3.73367452017986834486882804579, −2.58954535698109018311234781500, −1.92325686253028341240956056872, −0.29574091581702251528318757179, 0.792971381362917272931947788805, 2.31668643047815727870522719184, 3.16898121966913030585047533929, 4.09586541496072180862625666408, 5.47094845609024238215398754536, 5.74521764255956550861677182234, 6.70775477061188036023846313028, 7.54228171024679255469400081901, 7.934381338950786172514166853342, 8.963999523610793005419934787688

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