Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.935 + 0.354i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.76 − 3.04i)2-s + (−1.5 + 0.866i)3-s + (−4.19 − 7.26i)4-s + (−1.93 − 1.11i)5-s + 6.09i·6-s + (0.244 − 6.99i)7-s − 15.4·8-s + (1.5 − 2.59i)9-s + (−6.81 + 3.93i)10-s + (−1.29 − 2.24i)11-s + (12.5 + 7.26i)12-s + 11.5i·13-s + (−20.8 − 13.0i)14-s + 3.87·15-s + (−10.4 + 18.0i)16-s + (20.0 − 11.6i)17-s + ⋯
L(s)  = 1  + (0.880 − 1.52i)2-s + (−0.5 + 0.288i)3-s + (−1.04 − 1.81i)4-s + (−0.387 − 0.223i)5-s + 1.01i·6-s + (0.0348 − 0.999i)7-s − 1.93·8-s + (0.166 − 0.288i)9-s + (−0.681 + 0.393i)10-s + (−0.117 − 0.204i)11-s + (1.04 + 0.605i)12-s + 0.890i·13-s + (−1.49 − 0.932i)14-s + 0.258·15-s + (−0.652 + 1.12i)16-s + (1.18 − 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.935 + 0.354i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.935 + 0.354i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.278270 - 1.52065i\)
\(L(\frac12)\)  \(\approx\)  \(0.278270 - 1.52065i\)
\(L(2)\)   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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-0.244 + 6.99i)T \)
good2 \( 1 + (-1.76 + 3.04i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (1.29 + 2.24i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.5iT - 169T^{2} \)
17 \( 1 + (-20.0 + 11.6i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-25.9 - 14.9i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.5 + 30.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 24.4T + 841T^{2} \)
31 \( 1 + (32.4 - 18.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (12.8 - 22.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 3.71iT - 1.68e3T^{2} \)
43 \( 1 - 74.2T + 1.84e3T^{2} \)
47 \( 1 + (-2.92 - 1.68i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-20.0 - 34.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (42.7 - 24.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.765 + 0.441i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-32.5 - 56.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 86.0T + 5.04e3T^{2} \)
73 \( 1 + (53.3 - 30.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (13.7 - 23.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 + (56.5 + 32.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 42.2iT - 9.40e3T^{2} \)
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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

−12.77048891023475411981837952661, −11.95525748728497099976440785963, −11.14003998659759636921928591192, −10.29702448936839502694133609133, −9.315791051511566824700878401455, −7.33603135086065194023015476291, −5.48525760170800922757451307989, −4.39460168875906015145954876621, −3.33884572194486714846542008106, −1.05745444461378650393300367330, 3.43670449701224821037597431779, 5.29500350107647621641387515579, 5.73685808554343940944775344331, 7.25817770695117121682804876000, 7.87761730004735747886011074711, 9.331980513854808607889898432162, 11.19031818514638262719864789174, 12.35796004157610517039610728017, 13.00000800603543761078114587080, 14.19269889303317283667319962078

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