Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.206 - 0.978i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.167 − 0.167i)2-s + (−0.707 + 0.707i)3-s + 1.94i·4-s + 0.236i·6-s + (2.64 − 0.0627i)7-s + (0.658 + 0.658i)8-s − 1.00i·9-s + 3.98·11-s + (−1.37 − 1.37i)12-s + (−0.500 + 0.500i)13-s + (0.431 − 0.452i)14-s − 3.66·16-s + (1.67 + 1.67i)17-s + (−0.167 − 0.167i)18-s − 7.21·19-s + ⋯
L(s)  = 1  + (0.118 − 0.118i)2-s + (−0.408 + 0.408i)3-s + 0.972i·4-s + 0.0964i·6-s + (0.999 − 0.0237i)7-s + (0.232 + 0.232i)8-s − 0.333i·9-s + 1.20·11-s + (−0.396 − 0.396i)12-s + (−0.138 + 0.138i)13-s + (0.115 − 0.120i)14-s − 0.917·16-s + (0.407 + 0.407i)17-s + (−0.0393 − 0.0393i)18-s − 1.65·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.206 - 0.978i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.206 - 0.978i)\)
\(L(1)\)  \(\approx\)  \(1.15619 + 0.937532i\)
\(L(\frac12)\)  \(\approx\)  \(1.15619 + 0.937532i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 + 0.0627i)T \)
good2 \( 1 + (-0.167 + 0.167i)T - 2iT^{2} \)
11 \( 1 - 3.98T + 11T^{2} \)
13 \( 1 + (0.500 - 0.500i)T - 13iT^{2} \)
17 \( 1 + (-1.67 - 1.67i)T + 17iT^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + (-5.16 - 5.16i)T + 23iT^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 - 4.93iT - 31T^{2} \)
37 \( 1 + (0.292 - 0.292i)T - 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (3.65 + 3.65i)T + 43iT^{2} \)
47 \( 1 + (-0.305 - 0.305i)T + 47iT^{2} \)
53 \( 1 + (5.39 + 5.39i)T + 53iT^{2} \)
59 \( 1 - 6.10T + 59T^{2} \)
61 \( 1 - 7.11iT - 61T^{2} \)
67 \( 1 + (0.944 - 0.944i)T - 67iT^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (1.38 - 1.38i)T - 73iT^{2} \)
79 \( 1 + 8.64iT - 79T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 - 7.82T + 89T^{2} \)
97 \( 1 + (7.43 + 7.43i)T + 97iT^{2} \)
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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

−11.16759761775196453324635994748, −10.42166381740154282152972994389, −9.012882911889267374225751107820, −8.588698286928173394952055909407, −7.41151790124950242472893664951, −6.57961457658482298180001689590, −5.22021545799015036017413613523, −4.29752432917559528673179465751, −3.46056031560316024364367422331, −1.76244600401334390775556571350, 0.990411593222280616200215389380, 2.20487769300152039450162614917, 4.29224259474778683965178709482, 5.01135343645831204109165775044, 6.17811949663577628544107085074, 6.72693644705053578474769519989, 7.935769884750667501159680983758, 8.904115519266648139667858070478, 9.861788725625197625946233750195, 10.90171704266034654258527465163

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