Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.116 - 0.993i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + (−1 − 1.41i)3-s + 0.999·4-s + (−1.73 + 1.41i)5-s + (1.73 + 2.44i)6-s + (1 + 2.44i)7-s + 1.73·8-s + (−1.00 + 2.82i)9-s + (2.99 − 2.44i)10-s + 2.82i·11-s + (−0.999 − 1.41i)12-s − 4·13-s + (−1.73 − 4.24i)14-s + (3.73 + 1.03i)15-s − 5·16-s + 2.82i·17-s + ⋯
L(s)  = 1  − 1.22·2-s + (−0.577 − 0.816i)3-s + 0.499·4-s + (−0.774 + 0.632i)5-s + (0.707 + 0.999i)6-s + (0.377 + 0.925i)7-s + 0.612·8-s + (−0.333 + 0.942i)9-s + (0.948 − 0.774i)10-s + 0.852i·11-s + (−0.288 − 0.408i)12-s − 1.10·13-s + (−0.462 − 1.13i)14-s + (0.963 + 0.267i)15-s − 1.25·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.116 - 0.993i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (104, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.116 - 0.993i)$
$L(1)$  $\approx$  $0.176161 + 0.198079i$
$L(\frac12)$  $\approx$  $0.176161 + 0.198079i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (1 + 1.41i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
7 \( 1 + (-1 - 2.44i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 9.79iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 9.79iT - 61T^{2} \)
67 \( 1 + 4.89iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−14.16040645937814404352499902431, −12.55542785953301649178268002539, −11.87612198063923087218267320802, −10.83795825526564754182204473776, −9.813898458048051618481733890212, −8.336372474437866384699699670877, −7.64916586128995768211890223380, −6.60881536426890603723315013468, −4.84035481314464242271184792663, −2.13235648433281233133195991540, 0.47956781225255046972740175522, 4.01762483810657144674943936476, 5.11886651210099380061908742700, 7.16447954574929738174630481549, 8.187704968624703758502995805225, 9.254308968911655758612174884459, 10.19070171703711611628078833786, 11.13184856741464165409942417302, 11.94639858150923202707204186768, 13.51071119108233875096236476559

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