Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.400 + 0.916i$
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 + (2.23 − 1.41i)7-s − 8-s − 1.41i·11-s + 5.39·13-s + (−2.23 + 1.41i)14-s + 16-s − 2.23i·17-s + 1.30i·19-s + 1.41i·22-s − 23-s − 5.39·26-s + (2.23 − 1.41i)28-s − 9.24i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.845 − 0.534i)7-s − 0.353·8-s − 0.426i·11-s + 1.49·13-s + (−0.597 + 0.377i)14-s + 0.250·16-s − 0.542i·17-s + 0.300i·19-s + 0.301i·22-s − 0.208·23-s − 1.05·26-s + (0.422 − 0.267i)28-s − 1.71i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.400 + 0.916i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.400 + 0.916i)$
$L(1)$  $\approx$  $1.506887979$
$L(\frac12)$  $\approx$  $1.506887979$
$L(\frac{3}{2})$   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 \( 1 + (-2.23 + 1.41i)T \)
good11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 - 1.30iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 9.24iT - 29T^{2} \)
31 \( 1 + 8.56iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 - 4.08T + 41T^{2} \)
43 \( 1 - 6.41iT - 43T^{2} \)
47 \( 1 - 7.63iT - 47T^{2} \)
53 \( 1 - 6.07T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 5.39iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 4.34iT - 71T^{2} \)
73 \( 1 - 5.01T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 8.01iT - 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.305204230031972119231515587163, −8.037170477523292035786303089548, −7.28279984870819510406459061734, −6.20168892471255021340612162142, −5.82005755491265537459993612372, −4.53058625399700144637155248258, −3.85336897267693793237787857978, −2.72796970284675827340773650657, −1.59567589643450197558645628472, −0.66876321119493066531613599678, 1.21819689321840552581469706288, 1.93633265305773165420542068486, 3.12440634316891798124181334725, 4.06907688792391475370019832497, 5.14954079692918725464441484334, 5.80876814000058531693700558426, 6.73464609885703768456072381838, 7.37661171790404812779295917248, 8.408905125988366412537697382565, 8.644115147538370454275037459634

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