Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.391 + 0.920i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 3.41i·11-s + (−4 + 4i)13-s − 1.00·14-s − 1.00·16-s + (3.41 − 3.41i)17-s − 2.82i·19-s + (2.41 + 2.41i)22-s + (−0.828 − 0.828i)23-s + 5.65i·26-s + (−0.707 + 0.707i)28-s + 4.82·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 1.02i·11-s + (−1.10 + 1.10i)13-s − 0.267·14-s − 0.250·16-s + (0.828 − 0.828i)17-s − 0.648i·19-s + (0.514 + 0.514i)22-s + (−0.172 − 0.172i)23-s + 1.10i·26-s + (−0.133 + 0.133i)28-s + 0.896·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.391 + 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.391 + 0.920i)$
$L(1)$  $\approx$  $2.191756996$
$L(\frac12)$  $\approx$  $2.191756996$
$L(\frac{3}{2})$   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 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (4 - 4i)T - 13iT^{2} \)
17 \( 1 + (-3.41 + 3.41i)T - 17iT^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + (0.828 + 0.828i)T + 23iT^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-2.24 - 2.24i)T + 37iT^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 + (-8.07 + 8.07i)T - 43iT^{2} \)
47 \( 1 + (-0.757 + 0.757i)T - 47iT^{2} \)
53 \( 1 + (5.41 + 5.41i)T + 53iT^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 - 9.89T + 61T^{2} \)
67 \( 1 + (5.58 + 5.58i)T + 67iT^{2} \)
71 \( 1 - 6.82iT - 71T^{2} \)
73 \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \)
79 \( 1 + 5.65iT - 79T^{2} \)
83 \( 1 + (-4.82 - 4.82i)T + 83iT^{2} \)
89 \( 1 + 8.82T + 89T^{2} \)
97 \( 1 + (-7.41 - 7.41i)T + 97iT^{2} \)
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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

−8.703823732564818599949843278514, −7.57017194988745981861529337323, −6.99736594989227501228524020859, −6.36443404995698919683120781707, −5.14464916689989158402647429335, −4.68064805368551518299335524531, −3.91179107788775218385828811412, −2.74330350723390561401731630015, −2.12895179207047191317585245611, −0.71873921477873437996708461310, 0.963382300773776048038622673884, 2.67356263157165126108825506957, 3.17303382254405491354622127774, 4.23038619623491200332514704539, 5.09235137449360328006964745444, 5.96556388729189918560825686561, 6.21845187694929376322180975507, 7.40701075465569345137619375346, 8.107618497394323786690192501480, 8.421401744701951596377059128493

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