Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.0955 + 0.995i$
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 + (0.648 − 2.56i)7-s + 8-s − 1.29i·11-s − 3.13·13-s + (0.648 − 2.56i)14-s + 16-s − 5.53i·17-s + 7.37i·19-s − 1.29i·22-s + 1.83·23-s − 3.13·26-s + (0.648 − 2.56i)28-s − 1.83i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.245 − 0.969i)7-s + 0.353·8-s − 0.390i·11-s − 0.868·13-s + (0.173 − 0.685i)14-s + 0.250·16-s − 1.34i·17-s + 1.69i·19-s − 0.276i·22-s + 0.382·23-s − 0.613·26-s + (0.122 − 0.484i)28-s − 0.340i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0955 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0955 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0955 + 0.995i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.0955 + 0.995i)$
$L(1)$  $\approx$  $2.432385317$
$L(\frac12)$  $\approx$  $2.432385317$
$L(\frac{3}{2})$   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 \( 1 + (-0.648 + 2.56i)T \)
good11 \( 1 + 1.29iT - 11T^{2} \)
13 \( 1 + 3.13T + 13T^{2} \)
17 \( 1 + 5.53iT - 17T^{2} \)
19 \( 1 - 7.37iT - 19T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 + 1.83iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 + 3.13T + 41T^{2} \)
43 \( 1 + 3.53iT - 43T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 - 4.42T + 53T^{2} \)
59 \( 1 - 7.18T + 59T^{2} \)
61 \( 1 - 4.88iT - 61T^{2} \)
67 \( 1 + 9.79iT - 67T^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + 3.40T + 73T^{2} \)
79 \( 1 + 9.01T + 79T^{2} \)
83 \( 1 + 6.26iT - 83T^{2} \)
89 \( 1 + 7.94T + 89T^{2} \)
97 \( 1 + 8.09T + 97T^{2} \)
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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

−8.237966170505910174196266981578, −7.49548744485782349220569061004, −7.15411943736969473019869507468, −6.03890572602552242422539156764, −5.43399579006752365645245882127, −4.47857167567494955021794108951, −3.90760292369328577936566438318, −2.93556186395220294260344648642, −1.92438458810233149404694516545, −0.56035713599072284673937367531, 1.51468298139453488031707597266, 2.50788920139638131762570340519, 3.20587245135583765350188560769, 4.41577781559922645496494518742, 5.05069047959617342060514437252, 5.61459687910061527218558287040, 6.83481319028271357075636483866, 6.94396833885385829294129704589, 8.337639844839377222739961624631, 8.632084408201940679830593753236

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