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 + 5.65i·11-s + 4.47i·13-s + (−2.23 − 1.41i)14-s + 16-s + 3.16·17-s + 3.16i·19-s − 5.65·22-s + 4i·23-s − 4.47·26-s + (1.41 − 2.23i)28-s − 2.82i·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.534 + 0.845i)7-s − 0.353i·8-s + 1.70i·11-s + 1.24i·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s + 0.766·17-s + 0.725i·19-s − 1.20·22-s + 0.834i·23-s − 0.877·26-s + (0.267 − 0.422i)28-s − 0.525i·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  =  $(2,\ 3150,\ (\ :1/2),\ -0.924 + 0.381i)$
$L(1)$  $\approx$  $1.180540751$
$L(\frac12)$  $\approx$  $1.180540751$
$L(\frac{3}{2})$   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 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 3.16iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 1.41T + 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 9.48iT - 61T^{2} \)
67 \( 1 - 7.07T + 67T^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.347790483446802875585042874379, −8.150903898339289162127817745653, −7.61891739103387380771587696128, −6.80979243944588292044876742203, −6.15473782074428545645702734383, −5.42967170470950844594149937766, −4.51117455396822307117557610711, −3.84727952996583726920311026144, −2.56401406705641685592371569511, −1.61269610049448858986037593532, 0.41344192160225552189843759391, 1.14640946304978255392084492053, 2.97491059785818326077798251071, 3.11754358940727775660071153124, 4.15390353081677441799385799665, 5.14330091496514857869600555765, 5.92022351094780232746267307405, 6.68871857480235029663548533388, 7.73408387207304427069643172755, 8.303098308938658329021185111857

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