Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.233 + 0.972i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.54 − 1.54i)2-s + (1.73 + 0.00622i)3-s + 2.76i·4-s + (−2.66 − 2.68i)6-s + (−0.707 + 0.707i)7-s + (1.18 − 1.18i)8-s + (2.99 + 0.0215i)9-s − 3.38i·11-s + (−0.0172 + 4.79i)12-s + (0.206 + 0.206i)13-s + 2.18·14-s + 1.87·16-s + (−0.167 − 0.167i)17-s + (−4.59 − 4.66i)18-s − 5.31i·19-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s + (0.999 + 0.00359i)3-s + 1.38i·4-s + (−1.08 − 1.09i)6-s + (−0.267 + 0.267i)7-s + (0.419 − 0.419i)8-s + (0.999 + 0.00718i)9-s − 1.02i·11-s + (−0.00497 + 1.38i)12-s + (0.0573 + 0.0573i)13-s + 0.583·14-s + 0.467·16-s + (−0.0406 − 0.0406i)17-s + (−1.08 − 1.09i)18-s − 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.233 + 0.972i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (218, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.233 + 0.972i)\)
\(L(1)\)  \(\approx\)  \(0.691513 - 0.876996i\)
\(L(\frac12)\)  \(\approx\)  \(0.691513 - 0.876996i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.73 - 0.00622i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.54 + 1.54i)T + 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 + (0.167 + 0.167i)T + 17iT^{2} \)
19 \( 1 + 5.31iT - 19T^{2} \)
23 \( 1 + (-5.07 + 5.07i)T - 23iT^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 - 9.11T + 31T^{2} \)
37 \( 1 + (-5.27 + 5.27i)T - 37iT^{2} \)
41 \( 1 + 0.0314iT - 41T^{2} \)
43 \( 1 + (-3.76 - 3.76i)T + 43iT^{2} \)
47 \( 1 + (-3.56 - 3.56i)T + 47iT^{2} \)
53 \( 1 + (3.55 - 3.55i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 6.80T + 61T^{2} \)
67 \( 1 + (6.34 - 6.34i)T - 67iT^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (8.61 + 8.61i)T + 73iT^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + (3.88 - 3.88i)T - 83iT^{2} \)
89 \( 1 - 2.00T + 89T^{2} \)
97 \( 1 + (2.26 - 2.26i)T - 97iT^{2} \)
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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

−10.61569540360616486610838619407, −9.495774205794162945945465240037, −9.026570599561806058617475252906, −8.369837938946322236458560581653, −7.44531799936556615579034432840, −6.18319199208820336606329158996, −4.48787574779911140062019519737, −3.07623269960313593546898293099, −2.56611613960241575088882195394, −0.952697459981378890041056595491, 1.48357862972854387416017842213, 3.19447363552114762954626976877, 4.51133382122944267063599731889, 5.96756443332669802374390727947, 7.01240577053917730529835145228, 7.59658755235925573026231355817, 8.321611517754157193287656584895, 9.273786221423852841086624588150, 9.812304981368578216160759582968, 10.50678705497768124850711216382

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