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Properties

Degree 2
Conductor $ 5^{2} \cdot 7^{2} $
Sign $0.850 - 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·4-s − i·9-s + 2·11-s − 16-s + 2i·29-s + 36-s + 2i·44-s − i·64-s − 2·71-s − 2i·79-s − 81-s − 2i·99-s − 2i·109-s − 2·116-s + ⋯
L(s)  = 1  + i·4-s − i·9-s + 2·11-s − 16-s + 2i·29-s + 36-s + 2i·44-s − i·64-s − 2·71-s − 2i·79-s − 81-s − 2i·99-s − 2i·109-s − 2·116-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1225\)    =    \(5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.850 - 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{1225} (393, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1225,\ (\ :0),\ 0.850 - 0.525i)$
$L(\frac{1}{2})$  $\approx$  $1.152995143$
$L(\frac12)$  $\approx$  $1.152995143$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 - iT^{2} \)
3 \( 1 + iT^{2} \)
11 \( 1 - 2T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.756135287439598217163611926691, −8.927396946595284034690335471848, −8.706641390372644014055676846940, −7.36289440792070998940633942859, −6.79887453799926084924138946176, −6.05061216867643441238787335706, −4.60542854362445325517668098305, −3.76402609560450591347004108138, −3.13566014032118967832990293892, −1.48184834938861756288084290103, 1.29939866974928397743549768394, 2.34541662112690771483706212326, 3.95316090989407502307061453701, 4.68700314514062367333961334014, 5.76606802056393615800514373081, 6.40576646711482175871253270842, 7.26581164406950127297146803457, 8.330780849397573352821918570733, 9.202369470821391974638568171348, 9.808525302105704979753952440467

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