Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.447 - 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s i·7-s + i·8-s − 4·11-s − 6i·13-s − 14-s + 16-s + 4i·17-s − 6·19-s + 4i·22-s − 6·26-s + i·28-s + 6·29-s − 4·31-s i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 1.20·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s + 0.970i·17-s − 1.37·19-s + 0.852i·22-s − 1.17·26-s + 0.188i·28-s + 1.11·29-s − 0.718·31-s − 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.447 - 0.894i)$
$L(1)$  $\approx$  $0.5047815793$
$L(\frac12)$  $\approx$  $0.5047815793$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 6iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.663187105128396370217008490890, −8.160618847450276972319110530546, −7.61621120401563747472151358691, −6.43161845270593378323672486691, −5.67488746212090373125868442909, −4.86627943848617486282116948084, −4.05239657424506183186052804338, −3.06729337701200944141762454417, −2.39856718184895471042088853610, −1.07983564110204276269333971784, 0.16961349179264375423260645715, 1.95996452690452873532812760320, 2.81251756406472609130400871824, 4.16594551363586349195500193966, 4.68397690188515935167409193995, 5.61545149263463596047728862679, 6.28220007064314980836071518760, 7.14694670240559627966143056293, 7.61865278886682822730325139366, 8.728574922293480237348553011838

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