Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.254 + 0.966i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−1.93 + 1.80i)7-s + 8-s − 3.87i·11-s − 1.60·13-s + (−1.93 + 1.80i)14-s + 16-s − 8.11i·17-s + 2.63i·19-s − 3.87i·22-s + 5.47·23-s − 1.60·26-s + (−1.93 + 1.80i)28-s + 5.47i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.732 + 0.681i)7-s + 0.353·8-s − 1.16i·11-s − 0.445·13-s + (−0.517 + 0.481i)14-s + 0.250·16-s − 1.96i·17-s + 0.605i·19-s − 0.825i·22-s + 1.14·23-s − 0.314·26-s + (−0.366 + 0.340i)28-s + 1.01i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.254 + 0.966i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.254 + 0.966i)$
$L(1)$  $\approx$  $2.158785871$
$L(\frac12)$  $\approx$  $2.158785871$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (1.93 - 1.80i)T \)
good11 \( 1 + 3.87iT - 11T^{2} \)
13 \( 1 + 1.60T + 13T^{2} \)
17 \( 1 + 8.11iT - 17T^{2} \)
19 \( 1 - 2.63iT - 19T^{2} \)
23 \( 1 - 5.47T + 23T^{2} \)
29 \( 1 - 5.47iT - 29T^{2} \)
31 \( 1 - 3.73iT - 31T^{2} \)
37 \( 1 + 4.51iT - 37T^{2} \)
41 \( 1 + 1.60T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 + 2.26T + 53T^{2} \)
59 \( 1 + 4.61T + 59T^{2} \)
61 \( 1 + 11.8iT - 61T^{2} \)
67 \( 1 + 6.90iT - 67T^{2} \)
71 \( 1 + 2.63iT - 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 - 3.20iT - 83T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 + 8.68T + 97T^{2} \)
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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

−8.755330888268550120684744140990, −7.53561089714368482105030007568, −6.93112714161836962573004115022, −6.19381481933793898995378151911, −5.27534644090836637046615164074, −4.98086289277623844575653016572, −3.45997565510877550546380971304, −3.15404008744243660819514093101, −2.13415623329750287609864253305, −0.52444824243644774132262897061, 1.29817563836353451265011282374, 2.45725995683282654793768652390, 3.34783456709008293357639479233, 4.34929652012104093354322397673, 4.67174352543235743122995192541, 5.98690874817926186193061902270, 6.42119224604955308847526252258, 7.31264907046808807303689876575, 7.78380728106547524112998102443, 8.880578845338611870090868150532

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