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  :  $\chi_{105} (23, \cdot )$
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

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

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