Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.405 + 0.914i$
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 + (−2.27 + 1.35i)7-s i·8-s + 2.71i·11-s + 6.54i·13-s + (−1.35 − 2.27i)14-s + 16-s − 1.53·17-s − 2.30i·19-s − 2.71·22-s + 3.83i·23-s − 6.54·26-s + (2.27 − 1.35i)28-s − 3.83i·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.858 + 0.512i)7-s − 0.353i·8-s + 0.817i·11-s + 1.81i·13-s + (−0.362 − 0.607i)14-s + 0.250·16-s − 0.371·17-s − 0.528i·19-s − 0.577·22-s + 0.799i·23-s − 1.28·26-s + (0.429 − 0.256i)28-s − 0.711i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.405 + 0.914i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.405 + 0.914i)$
$L(1)$  $\approx$  $0.3811740276$
$L(\frac12)$  $\approx$  $0.3811740276$
$L(\frac{3}{2})$   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 \( 1 + (2.27 - 1.35i)T \)
good11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 - 6.54iT - 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 + 2.30iT - 19T^{2} \)
23 \( 1 - 3.83iT - 23T^{2} \)
29 \( 1 + 3.83iT - 29T^{2} \)
31 \( 1 + 3.25iT - 31T^{2} \)
37 \( 1 + 3.01T + 37T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 - 0.468T + 43T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 - 9.25iT - 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 4.78iT - 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 2.30iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−9.075915280690686580465296448079, −8.640870165489566632485427516220, −7.35800746058932671910307824408, −7.03912932225303191655368095366, −6.25525608669027424936171126656, −5.57047802680550372546300573911, −4.50981084447557890230102656544, −3.99876219675331723756603013257, −2.74184197014620646164422497747, −1.73607955160185689410687352916, 0.12878817138318499981805586129, 1.12061843740778628208891698941, 2.61777538240723846530650680493, 3.29866918555952496522794985029, 3.93051422437978095487167499826, 5.11369619577876174481102288874, 5.76733963735182311292784449402, 6.64343796702088489030052197270, 7.49011710639642934662094036458, 8.434285314051787939006170934166

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