Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.547 - 0.836i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + (−0.0951 + 2.64i)7-s + i·8-s + 5.28i·11-s − 2.19i·13-s + (2.64 + 0.0951i)14-s + 16-s − 1.04·17-s + 6.43i·19-s + 5.28·22-s − 7.47i·23-s − 2.19·26-s + (0.0951 − 2.64i)28-s + 7.47i·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.0359 + 0.999i)7-s + 0.353i·8-s + 1.59i·11-s − 0.607i·13-s + (0.706 + 0.0254i)14-s + 0.250·16-s − 0.253·17-s + 1.47i·19-s + 1.12·22-s − 1.55i·23-s − 0.429·26-s + (0.0179 − 0.499i)28-s + 1.38i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.547 - 0.836i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.547 - 0.836i)$
$L(1)$  $\approx$  $0.6365975010$
$L(\frac12)$  $\approx$  $0.6365975010$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.0951 - 2.64i)T \)
good11 \( 1 - 5.28iT - 11T^{2} \)
13 \( 1 + 2.19iT - 13T^{2} \)
17 \( 1 + 1.04T + 17T^{2} \)
19 \( 1 - 6.43iT - 19T^{2} \)
23 \( 1 + 7.47iT - 23T^{2} \)
29 \( 1 - 7.47iT - 29T^{2} \)
31 \( 1 + 9.09iT - 31T^{2} \)
37 \( 1 - 0.855T + 37T^{2} \)
41 \( 1 + 2.19T + 41T^{2} \)
43 \( 1 - 0.954T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 3.09iT - 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + 5.33T + 67T^{2} \)
71 \( 1 + 6.43iT - 71T^{2} \)
73 \( 1 + 4.57iT - 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 4.38T + 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 - 11.8iT - 97T^{2} \)
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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

−9.097417305828548760651132973326, −8.245493333362655473362396332674, −7.66862621559015938256391992977, −6.58762878306877140251035692084, −5.83021664595739277194153467766, −4.93521868377378313801878558992, −4.30630665653904665578194173778, −3.21153580476244868162254556021, −2.33137771344974794232911307186, −1.58940311492578859552958713557, 0.19873983003084499707183127570, 1.37632763665292059651761729974, 3.01653851854856646531370819900, 3.73642836469040030198739099510, 4.65740073921136099336954018022, 5.38818361919009060137774198582, 6.39392440730882804582665070598, 6.79925429399193409078747454876, 7.69027024983537908468194541993, 8.290255093040938746659901371111

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