Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.924 - 0.381i$
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 + (1.41 + 2.23i)7-s i·8-s + 1.41i·11-s + 0.926i·13-s + (−2.23 + 1.41i)14-s + 16-s + 2.23·17-s + 7.63i·19-s − 1.41·22-s i·23-s − 0.926·26-s + (−1.41 − 2.23i)28-s + 0.757i·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.534 + 0.845i)7-s − 0.353i·8-s + 0.426i·11-s + 0.256i·13-s + (−0.597 + 0.377i)14-s + 0.250·16-s + 0.542·17-s + 1.75i·19-s − 0.301·22-s − 0.208i·23-s − 0.181·26-s + (−0.267 − 0.422i)28-s + 0.140i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.924 - 0.381i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.924 - 0.381i)$
$L(1)$  $\approx$  $1.512139702$
$L(\frac12)$  $\approx$  $1.512139702$
$L(\frac{3}{2})$   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 \( 1 + (-1.41 - 2.23i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 0.926iT - 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 - 7.63iT - 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 0.757iT - 29T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 - 8.56T + 41T^{2} \)
43 \( 1 + 3.58T + 43T^{2} \)
47 \( 1 + 1.30T + 47T^{2} \)
53 \( 1 + 8.07iT - 53T^{2} \)
59 \( 1 - 7.25T + 59T^{2} \)
61 \( 1 + 0.926iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 - 13.9iT - 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 2.61T + 89T^{2} \)
97 \( 1 - 0.542iT - 97T^{2} \)
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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

−8.736907036853967628001283303190, −8.256938466278184018640801626725, −7.58149295525803678366181279442, −6.78085165311332912105752972782, −5.81641246244526399315888749616, −5.46863098867844809702236396617, −4.46186289239254118076880504362, −3.68498636727086467168040396913, −2.43888306816864550573205535280, −1.39831453302952083460859883680, 0.49811295265645591645045166457, 1.47753366969311752049558230590, 2.70902233278820740219729840214, 3.47714639900838048032494317869, 4.44388347225842337958915556080, 5.03474084310604522010238671613, 5.97834718570700753061868131111, 7.03785532770434825504059405698, 7.62119208014701831799845082741, 8.494677155459885479383637906151

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