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 + 0.926·13-s + (−2.23 + 1.41i)14-s + 16-s − 2.23i·17-s − 7.63i·19-s − 1.41i·22-s + 23-s + 0.926·26-s + (−2.23 + 1.41i)28-s + 0.757i·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 + 0.256·13-s + (−0.597 + 0.377i)14-s + 0.250·16-s − 0.542i·17-s − 1.75i·19-s − 0.301i·22-s + 0.208·23-s + 0.181·26-s + (−0.422 + 0.267i)28-s + 0.140i·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  =  $(2,\ 3150,\ (\ :1/2),\ 0.400 + 0.916i)$
$L(1)$  $\approx$  $2.270048407$
$L(\frac12)$  $\approx$  $2.270048407$
$L(\frac{3}{2})$   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 \( 1 - 0.926T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + 7.63iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - 0.757iT - 29T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 + 8.56T + 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 - 1.30iT - 47T^{2} \)
53 \( 1 - 8.07T + 53T^{2} \)
59 \( 1 - 7.25T + 59T^{2} \)
61 \( 1 + 0.926iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + 2.61T + 89T^{2} \)
97 \( 1 + 0.542T + 97T^{2} \)
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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

−8.673861883417906527688942527893, −7.63170076015691668174726479926, −6.80520264011440535696294946420, −6.31245951287508781407480920155, −5.41569187689051983483339597786, −4.79226887825033466898857391496, −3.70871632664595425332734892366, −2.97936886245568907935143633043, −2.20759258051920248665733107007, −0.56370170012592663241497367554, 1.26112194292300668730058430506, 2.38274476121712974568918724487, 3.63716920881122647887215034727, 3.81976549758620073170954285807, 5.00896754255527842359315829695, 5.77630377018373568345561127385, 6.55880208339295172921104501797, 7.08543651357902144225111212862, 8.008426221944900124120170710397, 8.691638802404206357627796908870

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