Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.327 + 0.944i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.347 + 0.347i)2-s + (−1.72 − 0.176i)3-s − 1.75i·4-s + (−0.536 − 0.659i)6-s + (0.707 − 0.707i)7-s + (1.30 − 1.30i)8-s + (2.93 + 0.607i)9-s + 2.67i·11-s + (−0.310 + 3.03i)12-s + (−2.14 − 2.14i)13-s + 0.490·14-s − 2.61·16-s + (−3.26 − 3.26i)17-s + (0.808 + 1.23i)18-s − 5.24i·19-s + ⋯
L(s)  = 1  + (0.245 + 0.245i)2-s + (−0.994 − 0.101i)3-s − 0.879i·4-s + (−0.219 − 0.269i)6-s + (0.267 − 0.267i)7-s + (0.461 − 0.461i)8-s + (0.979 + 0.202i)9-s + 0.805i·11-s + (−0.0895 + 0.874i)12-s + (−0.596 − 0.596i)13-s + 0.131·14-s − 0.653·16-s + (−0.792 − 0.792i)17-s + (0.190 + 0.290i)18-s − 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.327 + 0.944i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (218, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.327 + 0.944i)\)
\(L(1)\)  \(\approx\)  \(0.533244 - 0.749347i\)
\(L(\frac12)\)  \(\approx\)  \(0.533244 - 0.749347i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.72 + 0.176i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-0.347 - 0.347i)T + 2iT^{2} \)
11 \( 1 - 2.67iT - 11T^{2} \)
13 \( 1 + (2.14 + 2.14i)T + 13iT^{2} \)
17 \( 1 + (3.26 + 3.26i)T + 17iT^{2} \)
19 \( 1 + 5.24iT - 19T^{2} \)
23 \( 1 + (-2.54 + 2.54i)T - 23iT^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + (-2.14 + 2.14i)T - 37iT^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (0.759 + 0.759i)T + 43iT^{2} \)
47 \( 1 + (-7.66 - 7.66i)T + 47iT^{2} \)
53 \( 1 + (4.43 - 4.43i)T - 53iT^{2} \)
59 \( 1 - 0.159T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 + (-5.41 + 5.41i)T - 67iT^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (4.16 + 4.16i)T + 73iT^{2} \)
79 \( 1 + 3.89iT - 79T^{2} \)
83 \( 1 + (4.03 - 4.03i)T - 83iT^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 + (-1.86 + 1.86i)T - 97iT^{2} \)
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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

−10.81742048149702737117165560415, −9.844830697414123543057746600438, −9.082607893650920605144651279795, −7.23132056784694255127824634204, −7.09217497252076894725006834186, −5.81159669442540735443873759371, −4.98762238199600033402808876798, −4.37063446287184218478206340336, −2.18484985907224781962423762043, −0.55918542433165310246539073764, 1.87622861357476000252216729477, 3.51131354909262729064437999301, 4.43576385247174634111076444054, 5.46808228550363965027983182695, 6.47532943887545781223269743361, 7.48134438462922091044711635831, 8.405236559073647535749115670175, 9.408603334460123404323237445435, 10.54864645389173696611523755115, 11.38975940291679865179203970645

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