Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.968 - 0.249i$
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 + (−2.24 + 1.39i)7-s − 0.999i·8-s + (−1.37 − 0.796i)11-s − 0.925i·13-s + (−1.24 + 2.33i)14-s + (−0.5 − 0.866i)16-s + (−1.97 + 3.42i)17-s + (−0.541 + 0.312i)19-s − 1.59·22-s + (6.74 − 3.89i)23-s + (−0.462 − 0.801i)26-s + (0.0890 + 2.64i)28-s + 9.34i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.848 + 0.528i)7-s − 0.353i·8-s + (−0.415 − 0.240i)11-s − 0.256i·13-s + (−0.332 + 0.623i)14-s + (−0.125 − 0.216i)16-s + (−0.479 + 0.831i)17-s + (−0.124 + 0.0717i)19-s − 0.339·22-s + (1.40 − 0.811i)23-s + (−0.0907 − 0.157i)26-s + (0.0168 + 0.499i)28-s + 1.73i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.968 - 0.249i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.968 - 0.249i)$
$L(1)$  $\approx$  $2.150749159$
$L(\frac12)$  $\approx$  $2.150749159$
$L(\frac{3}{2})$   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 \( 1 + (2.24 - 1.39i)T \)
good11 \( 1 + (1.37 + 0.796i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.925iT - 13T^{2} \)
17 \( 1 + (1.97 - 3.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.541 - 0.312i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.74 + 3.89i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.34iT - 29T^{2} \)
31 \( 1 + (-8.94 - 5.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.213 + 0.369i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + (-1.39 - 2.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.90 - 1.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.10 + 5.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.52 + 5.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.178 + 0.308i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.07iT - 71T^{2} \)
73 \( 1 + (-5.91 - 3.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.52 - 7.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.809T + 83T^{2} \)
89 \( 1 + (-2.00 - 3.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.87iT - 97T^{2} \)
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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

−8.746584540253229429070045844716, −8.111297892281781418989404858658, −6.72094801301373923685374736206, −6.62882854472239882354271251513, −5.49573452039742254727539552304, −4.96180856952359243598662433829, −3.89556076429653821422504099706, −3.04217735664856867669342542242, −2.43000844973438054229152241830, −1.01569803865670054050694218676, 0.63690809874053381915225953010, 2.35484410218079165898300266741, 3.07436848386267280163373797272, 4.10095142768234344523656879934, 4.69883287122857538963606073075, 5.65961251647444780290611775031, 6.41738404601066344813532312170, 7.08643703744519589585621386440, 7.65373419758992484867074988557, 8.556267554238101116837958214206

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