Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.999 - 0.0304i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.17 + 0.582i)2-s + (−0.245 − 1.71i)3-s + (2.64 + 1.52i)4-s + (−1.39 + 1.74i)5-s + (0.465 − 3.86i)6-s + (−2.38 − 1.15i)7-s + (1.68 + 1.68i)8-s + (−2.87 + 0.840i)9-s + (−4.05 + 2.97i)10-s + (3.88 + 2.24i)11-s + (1.97 − 4.91i)12-s + (−1.08 + 1.08i)13-s + (−4.50 − 3.88i)14-s + (3.33 + 1.96i)15-s + (−0.381 − 0.660i)16-s + (−0.548 − 2.04i)17-s + ⋯
L(s)  = 1  + (1.53 + 0.411i)2-s + (−0.141 − 0.989i)3-s + (1.32 + 0.764i)4-s + (−0.625 + 0.780i)5-s + (0.189 − 1.57i)6-s + (−0.900 − 0.435i)7-s + (0.595 + 0.595i)8-s + (−0.959 + 0.280i)9-s + (−1.28 + 0.941i)10-s + (1.17 + 0.676i)11-s + (0.569 − 1.41i)12-s + (−0.300 + 0.300i)13-s + (−1.20 − 1.03i)14-s + (0.861 + 0.508i)15-s + (−0.0952 − 0.165i)16-s + (−0.133 − 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.999 - 0.0304i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.999 - 0.0304i)$
$L(1)$  $\approx$  $1.77724 + 0.0270975i$
$L(\frac12)$  $\approx$  $1.77724 + 0.0270975i$
$L(\frac{3}{2})$   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 \( 1 + (0.245 + 1.71i)T \)
5 \( 1 + (1.39 - 1.74i)T \)
7 \( 1 + (2.38 + 1.15i)T \)
good2 \( 1 + (-2.17 - 0.582i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-3.88 - 2.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.08 - 1.08i)T - 13iT^{2} \)
17 \( 1 + (0.548 + 2.04i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.66 + 2.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.840 - 3.13i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.69T + 29T^{2} \)
31 \( 1 + (-0.530 + 0.918i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 + 5.75i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 + 2.00i)T - 43iT^{2} \)
47 \( 1 + (-5.10 - 1.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.34 - 2.23i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.35 - 4.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.88 - 6.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.569 + 0.152i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (-1.13 - 4.22i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.78 - 3.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + (1.75 + 3.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.60 + 5.60i)T + 97iT^{2} \)
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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

−13.87552024674198393180164613483, −12.85779835183119662953609040219, −11.98782639543457670639294314889, −11.32722698699825928012973866601, −9.484988583525938544434189258813, −7.32458571187296670885860637271, −6.99437789411346467104041271805, −5.93618407899525473248575853656, −4.19285726630879414488112600269, −2.92526259499093625769628058426, 3.26257590060564266961992521243, 4.09749620859046307020401784457, 5.33179040698267009616339620043, 6.31327462721956530965088084939, 8.550855158235659709138410047521, 9.610318214848570576615933925396, 11.03311342381299490137523336792, 11.96793848185378110722648933532, 12.53000470174176973412340263550, 13.76903129923835160832066857567

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