Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.990 - 0.138i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 + 0.340i)2-s + (1.71 − 0.224i)3-s + (−0.236 − 0.136i)4-s + (−2.23 + 0.155i)5-s + (2.25 + 0.299i)6-s + (1.25 + 2.32i)7-s + (−2.11 − 2.11i)8-s + (2.89 − 0.769i)9-s + (−2.88 − 0.560i)10-s + (−3.38 − 1.95i)11-s + (−0.436 − 0.181i)12-s + (−1.56 + 1.56i)13-s + (0.807 + 3.38i)14-s + (−3.79 + 0.767i)15-s + (−1.69 − 2.92i)16-s + (0.693 + 2.58i)17-s + ⋯
L(s)  = 1  + (0.897 + 0.240i)2-s + (0.991 − 0.129i)3-s + (−0.118 − 0.0681i)4-s + (−0.997 + 0.0697i)5-s + (0.921 + 0.122i)6-s + (0.476 + 0.879i)7-s + (−0.746 − 0.746i)8-s + (0.966 − 0.256i)9-s + (−0.912 − 0.177i)10-s + (−1.01 − 0.588i)11-s + (−0.125 − 0.0523i)12-s + (−0.434 + 0.434i)13-s + (0.215 + 0.903i)14-s + (−0.980 + 0.198i)15-s + (−0.422 − 0.731i)16-s + (0.168 + 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.990 - 0.138i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.990 - 0.138i)$
$L(1)$  $\approx$  $1.60409 + 0.111403i$
$L(\frac12)$  $\approx$  $1.60409 + 0.111403i$
$L(\frac{3}{2})$   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 \( 1 + (-1.71 + 0.224i)T \)
5 \( 1 + (2.23 - 0.155i)T \)
7 \( 1 + (-1.25 - 2.32i)T \)
good2 \( 1 + (-1.26 - 0.340i)T + (1.73 + i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 - 1.56i)T - 13iT^{2} \)
17 \( 1 + (-0.693 - 2.58i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.638 + 2.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.513T + 29T^{2} \)
31 \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.77 + 6.60i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 + 7.60i)T - 43iT^{2} \)
47 \( 1 + (-5.10 - 1.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.85 - 0.498i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.259 + 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.74 - 2.34i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (0.749 + 2.79i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.37 - 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.16 - 9.16i)T + 83iT^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.81 + 6.81i)T + 97iT^{2} \)
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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

−14.04476911730175864701895066607, −12.78364656720614120043269657347, −12.22277915603689162951362519599, −10.73063916555009095615437961060, −9.151963358458087518970385244433, −8.324786303933787469103256224756, −7.15035430134809741173405557082, −5.46523422099078578061462401717, −4.21473173895594169618119723078, −2.89200156503303298503850693861, 2.90332607184030009539581897638, 4.12153798765723412104267812638, 4.99628925925173155759852912325, 7.53246715495307100045275636423, 7.931836715762474889750600424019, 9.410263205315957611564649382011, 10.71732685239381283729222942557, 11.93832364132465556409254539158, 12.96719708733341324247364503456, 13.64913726746391894423376119016

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