Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.913 + 0.406i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.907 + 0.243i)2-s + (−0.315 − 1.70i)3-s + (−0.967 + 0.558i)4-s + (−2.12 − 0.695i)5-s + (0.700 + 1.46i)6-s + (−2.64 − 0.0144i)7-s + (2.07 − 2.07i)8-s + (−2.80 + 1.07i)9-s + (2.09 + 0.114i)10-s + (−0.630 + 0.363i)11-s + (1.25 + 1.47i)12-s + (−1.44 − 1.44i)13-s + (2.40 − 0.630i)14-s + (−0.515 + 3.83i)15-s + (−0.257 + 0.446i)16-s + (1.90 − 7.09i)17-s + ⋯
L(s)  = 1  + (−0.641 + 0.171i)2-s + (−0.182 − 0.983i)3-s + (−0.483 + 0.279i)4-s + (−0.950 − 0.311i)5-s + (0.285 + 0.599i)6-s + (−0.999 − 0.00544i)7-s + (0.732 − 0.732i)8-s + (−0.933 + 0.357i)9-s + (0.663 + 0.0363i)10-s + (−0.189 + 0.109i)11-s + (0.362 + 0.425i)12-s + (−0.400 − 0.400i)13-s + (0.642 − 0.168i)14-s + (−0.133 + 0.991i)15-s + (−0.0644 + 0.111i)16-s + (0.460 − 1.71i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.913 + 0.406i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (23, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.913 + 0.406i)$
$L(1)$  $\approx$  $0.0480995 - 0.226720i$
$L(\frac12)$  $\approx$  $0.0480995 - 0.226720i$
$L(\frac{3}{2})$   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 \( 1 + (0.315 + 1.70i)T \)
5 \( 1 + (2.12 + 0.695i)T \)
7 \( 1 + (2.64 + 0.0144i)T \)
good2 \( 1 + (0.907 - 0.243i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.630 - 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 + 1.44i)T + 13iT^{2} \)
17 \( 1 + (-1.90 + 7.09i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.664 - 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.840 - 3.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.63 + 6.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 + 5.15i)T + 43iT^{2} \)
47 \( 1 + (6.79 - 1.82i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.26 + 1.41i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.807 + 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.90 + 1.84i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.06iT - 71T^{2} \)
73 \( 1 + (4.08 - 15.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.83 - 1.83i)T - 83iT^{2} \)
89 \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.62 + 5.62i)T - 97iT^{2} \)
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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

−13.06242183593282653023884684214, −12.44786031115851504753505492986, −11.41772825595437943527039866667, −9.826477280089150003082574258138, −8.819485477302651372097285436129, −7.61065296579138581281409204746, −7.10655572312070438381036716860, −5.21528112082486383027274294198, −3.33933860931327670379555380019, −0.33149081114965854031648209193, 3.43927710989046958441187004643, 4.67767214434778840890267072660, 6.25777017293679745743225331913, 8.004078040353215822389478885290, 9.006502445152273961977772482974, 10.06515475748515033271800068235, 10.68851160962053435269015230781, 11.84904773007001102844367343294, 13.12719217628758346557776855059, 14.60297831345143648037198391508

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