Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.894 + 0.447i$
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 + i·7-s i·8-s + 5·11-s − 6i·13-s − 14-s + 16-s i·17-s + 3·19-s + 5i·22-s + 6·26-s i·28-s − 6·29-s − 4·31-s + i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 1.50·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s − 0.242i·17-s + 0.688·19-s + 1.06i·22-s + 1.17·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s + 0.176i·32-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.894 + 0.447i)$
$L(1)$  $\approx$  $1.567479516$
$L(\frac12)$  $\approx$  $1.567479516$
$L(\frac{3}{2})$   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 \( 1 - 5T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
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

−8.647550035908286158871184300160, −7.76603679378902413954779017383, −7.18559469412112112694076262632, −6.35628908001268040720223029535, −5.56940118472467535869881663402, −5.08876612674051165008006426965, −3.82591323704486498966740123322, −3.29831504988979286938665182240, −1.85702953716993131741810715062, −0.52581178960635267967458547310, 1.26151310040410465549325609124, 1.89161967619458329935438229526, 3.27044647177164368629895254730, 3.97859582970042917261455859819, 4.57512033006114539987923451965, 5.63126177784664005361074925686, 6.67356107790910299249991892052, 7.04153297606879144723029512768, 8.193752853026428756146186289973, 8.940999227549445112879363638634

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