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 − 2.14i·11-s + (−2.16 + 2.16i)13-s − 1.00·14-s − 1.00·16-s + (−4.46 + 4.46i)17-s + (1.51 + 1.51i)22-s + (−6.32 − 6.32i)23-s − 3.05i·26-s + (0.707 − 0.707i)28-s + 8.20·29-s − 1.85·31-s + (0.707 − 0.707i)32-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 − 0.647i·11-s + (−0.599 + 0.599i)13-s − 0.267·14-s − 0.250·16-s + (−1.08 + 1.08i)17-s + (0.323 + 0.323i)22-s + (−1.31 − 1.31i)23-s − 0.599i·26-s + (0.133 − 0.133i)28-s + 1.52·29-s − 0.332·31-s + (0.125 − 0.125i)32-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  =  $(2,\ 3150,\ (\ :1/2),\ 0.391 + 0.920i)$
$L(1)$  $\approx$  $0.7676453985$
$L(\frac12)$  $\approx$  $0.7676453985$
$L(\frac{3}{2})$   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 \( 1 + 2.14iT - 11T^{2} \)
13 \( 1 + (2.16 - 2.16i)T - 13iT^{2} \)
17 \( 1 + (4.46 - 4.46i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (6.32 + 6.32i)T + 23iT^{2} \)
29 \( 1 - 8.20T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 + (-5.13 - 5.13i)T + 37iT^{2} \)
41 \( 1 + 7.56iT - 41T^{2} \)
43 \( 1 + (-7.84 + 7.84i)T - 43iT^{2} \)
47 \( 1 + (-5.01 + 5.01i)T - 47iT^{2} \)
53 \( 1 + (2.35 + 2.35i)T + 53iT^{2} \)
59 \( 1 + 9.35T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 + (-3.84 - 3.84i)T + 67iT^{2} \)
71 \( 1 + 0.420iT - 71T^{2} \)
73 \( 1 + (-2.63 + 2.63i)T - 73iT^{2} \)
79 \( 1 + 5.23iT - 79T^{2} \)
83 \( 1 + (10.6 + 10.6i)T + 83iT^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (-0.606 - 0.606i)T + 97iT^{2} \)
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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

−8.616723594942762843186271792580, −7.942338124934694319187804229996, −7.01462056932008710475666733605, −6.31317380764504775666622119874, −5.76527311964653184566595142571, −4.64163351070729919111427198570, −4.07704195422868462723287230891, −2.62708604389060384656534928051, −1.81399755582543843464312765105, −0.30817707298343377363356954312, 1.07821240856353839095209798899, 2.26973901088499906145189652975, 2.96337884542381945494313459814, 4.22644767527350789347575197723, 4.71594262316808421493883025014, 5.80646673166061180698410007875, 6.75075108635863548866149111893, 7.64332222252230170310648940918, 7.88667137361982322994137814982, 9.004116879620445656967515137757

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