Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.376 - 0.926i$
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 + (−1.35 + 2.27i)7-s + 8-s + 2.71i·11-s + 6.54·13-s + (−1.35 + 2.27i)14-s + 16-s − 1.53i·17-s − 2.30i·19-s + 2.71i·22-s − 3.83·23-s + 6.54·26-s + (−1.35 + 2.27i)28-s + 3.83i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.512 + 0.858i)7-s + 0.353·8-s + 0.817i·11-s + 1.81·13-s + (−0.362 + 0.607i)14-s + 0.250·16-s − 0.371i·17-s − 0.528i·19-s + 0.577i·22-s − 0.799·23-s + 1.28·26-s + (−0.256 + 0.429i)28-s + 0.711i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.376 - 0.926i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.376 - 0.926i)$
$L(1)$  $\approx$  $2.807999840$
$L(\frac12)$  $\approx$  $2.807999840$
$L(\frac{3}{2})$   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 \( 1 + (1.35 - 2.27i)T \)
good11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 - 6.54T + 13T^{2} \)
17 \( 1 + 1.53iT - 17T^{2} \)
19 \( 1 + 2.30iT - 19T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 - 3.25iT - 31T^{2} \)
37 \( 1 - 3.01iT - 37T^{2} \)
41 \( 1 - 6.54T + 41T^{2} \)
43 \( 1 - 0.468iT - 43T^{2} \)
47 \( 1 - 9.11iT - 47T^{2} \)
53 \( 1 + 9.25T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 4.78iT - 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 - 2.30iT - 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−8.804759828811386859174407724042, −8.107424823657408666562909426030, −7.10673551538897910930360362607, −6.39944781051027942938558498505, −5.83671051836349931248796481391, −5.00508499412957755895649895682, −4.12169505726356963852019908045, −3.28644459541553577125614612881, −2.44976105679750672904638654271, −1.34046830362660852855484600243, 0.71327364779792219196678408079, 1.90228104373557951356908323115, 3.28173445954675237586845274793, 3.76260729142084422505102420160, 4.41895168983808905544319514593, 5.87062005197135806264266638109, 5.97795524489540186838032887650, 6.84456776846451792299829205708, 7.82397955952932204557991294442, 8.354184945502779885956672617668

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