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  :  $\chi_{105} (23, \cdot )$
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

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

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