Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.919 - 0.391i$
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 + (−1.59 + 0.664i)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 + (2.11 − 2.12i)9-s + (−2.88 − 0.560i)10-s + (3.38 + 1.95i)11-s + (0.468 + 0.0611i)12-s + (−1.56 + 1.56i)13-s + (−0.807 − 3.38i)14-s + (−3.46 + 1.73i)15-s + (−1.69 − 2.92i)16-s + (−0.693 − 2.58i)17-s + ⋯
L(s)  = 1  + (−0.897 − 0.240i)2-s + (−0.923 + 0.383i)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 + (0.705 − 0.708i)9-s + (−0.912 − 0.177i)10-s + (1.01 + 0.588i)11-s + (0.135 + 0.0176i)12-s + (−0.434 + 0.434i)13-s + (−0.215 − 0.903i)14-s + (−0.894 + 0.447i)15-s + (−0.422 − 0.731i)16-s + (−0.168 − 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.919 - 0.391i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.919 - 0.391i)$
$L(1)$  $\approx$  $0.572052 + 0.116772i$
$L(\frac12)$  $\approx$  $0.572052 + 0.116772i$
$L(\frac{3}{2})$   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 \( 1 + (1.59 - 0.664i)T \)
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

−13.96956584825560252787638841650, −12.44936103572861679069922631887, −11.53331221984595996746831773782, −10.50805149112917444510601830433, −9.347623361032583298565125613893, −9.124047381503724995983117002220, −7.11672428074250424854903028499, −5.65823479227484294269606180645, −4.72969088619154278259603818389, −1.73904343822875846850639853300, 1.24324151723839757288615306673, 4.36729678002318017081788846191, 5.95298201019401092014987176496, 7.04958905861857775820358890950, 8.145833308543413029373144095922, 9.540267385189872841488655235290, 10.37958817601971674607072615681, 11.32482102268712550650756418832, 12.79677072743249089405635632074, 13.54885294685119897336684567112

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