Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.998 - 0.0618i$
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 + 5.26i·11-s + (3.16 − 3.16i)13-s + 1.00·14-s − 1.00·16-s + (−3.05 + 3.05i)17-s + (3.72 + 3.72i)22-s + (−4.32 − 4.32i)23-s − 4.46i·26-s + (0.707 − 0.707i)28-s + 9.96·29-s + 1.26·31-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 + 1.58i·11-s + (0.876 − 0.876i)13-s + 0.267·14-s − 0.250·16-s + (−0.741 + 0.741i)17-s + (0.793 + 0.793i)22-s + (−0.901 − 0.901i)23-s − 0.876i·26-s + (0.133 − 0.133i)28-s + 1.84·29-s + 0.227·31-s + (−0.125 + 0.125i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.998 - 0.0618i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.998 - 0.0618i)$
$L(1)$  $\approx$  $2.455910178$
$L(\frac12)$  $\approx$  $2.455910178$
$L(\frac{3}{2})$   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 \( 1 - 5.26iT - 11T^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \)
17 \( 1 + (3.05 - 3.05i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (4.32 + 4.32i)T + 23iT^{2} \)
29 \( 1 - 9.96T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (-2.93 - 2.93i)T + 37iT^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 + (0.597 - 0.597i)T - 43iT^{2} \)
47 \( 1 + (-3.42 + 3.42i)T - 47iT^{2} \)
53 \( 1 + (-9.88 - 9.88i)T + 53iT^{2} \)
59 \( 1 - 3.12T + 59T^{2} \)
61 \( 1 - 3.05T + 61T^{2} \)
67 \( 1 + (4.59 + 4.59i)T + 67iT^{2} \)
71 \( 1 - 9.23iT - 71T^{2} \)
73 \( 1 + (10.2 - 10.2i)T - 73iT^{2} \)
79 \( 1 - 3.57iT - 79T^{2} \)
83 \( 1 + (-6.21 - 6.21i)T + 83iT^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 + (-4.63 - 4.63i)T + 97iT^{2} \)
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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

−8.525863513775470649562124668020, −8.181079741040241093462864094932, −7.02908610534415542727532875507, −6.32608715796797043248598156943, −5.60676110266927252051899082650, −4.48412959469584296031065539378, −4.28313121622641622475991583056, −2.94636800630101030530047911341, −2.19520962180976842924003113648, −1.13309695910282967825420048890, 0.73100880806499381951908212707, 2.17606727749768082125180184637, 3.31902969878940883950723883418, 3.99607377658258471173073224776, 4.82384710225471684365453165851, 5.78244369138691479107711376021, 6.29707176780921845658479798212, 7.07787087976538763520372917699, 7.88856540504639974882037268790, 8.739546958810518605974732151823

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