Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.447 - 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s i·7-s + i·8-s + 2i·13-s − 14-s + 16-s − 6i·17-s − 8·19-s + 2·26-s + i·28-s + 6·29-s − 4·31-s i·32-s − 6·34-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 1.45i·17-s − 1.83·19-s + 0.392·26-s + 0.188i·28-s + 1.11·29-s − 0.718·31-s − 0.176i·32-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.447 - 0.894i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.339330707143783256402445841106, −7.46099889733317650240924556944, −6.68710253522566000263131383101, −5.90735893405611094874501195460, −4.64674453580542058632471879852, −4.43238676642480667306255207838, −3.22369915290844804845614088379, −2.43461805855537396249455206233, −1.34327336253784489312356493318, 0, 1.65176570952002498107029789879, 2.77879340538732471404902232733, 3.95157743770693289972200245937, 4.53507047752588653882541464811, 5.67703750528507140179191063606, 6.08766621867549647746327852455, 6.85961694364785670576843052593, 7.79856665122842137546509773647, 8.394663982777576601807785052690

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