Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.575 - 0.817i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 1.67i)2-s + (−2.55 − 1.56i)3-s + 1.58i·4-s + (0.529 − 4.97i)5-s + (1.66 + 6.89i)6-s + (−1.73 − 6.78i)7-s + (−4.03 + 4.03i)8-s + (4.10 + 8.01i)9-s + (−9.19 + 7.42i)10-s + 4.41i·11-s + (2.48 − 4.06i)12-s + (−1.62 + 1.62i)13-s + (−8.43 + 14.2i)14-s + (−9.13 + 11.8i)15-s + 19.8·16-s + (−13.9 + 13.9i)17-s + ⋯
L(s)  = 1  + (−0.835 − 0.835i)2-s + (−0.853 − 0.521i)3-s + 0.397i·4-s + (0.105 − 0.994i)5-s + (0.277 + 1.14i)6-s + (−0.247 − 0.968i)7-s + (−0.503 + 0.503i)8-s + (0.455 + 0.890i)9-s + (−0.919 + 0.742i)10-s + 0.401i·11-s + (0.207 − 0.338i)12-s + (−0.124 + 0.124i)13-s + (−0.602 + 1.01i)14-s + (−0.609 + 0.793i)15-s + 1.23·16-s + (−0.819 + 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.575 - 0.817i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.575 - 0.817i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.160985 + 0.310044i\)
\(L(\frac12)\)  \(\approx\)  \(0.160985 + 0.310044i\)
\(L(2)\)   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 \( 1 + (2.55 + 1.56i)T \)
5 \( 1 + (-0.529 + 4.97i)T \)
7 \( 1 + (1.73 + 6.78i)T \)
good2 \( 1 + (1.67 + 1.67i)T + 4iT^{2} \)
11 \( 1 - 4.41iT - 121T^{2} \)
13 \( 1 + (1.62 - 1.62i)T - 169iT^{2} \)
17 \( 1 + (13.9 - 13.9i)T - 289iT^{2} \)
19 \( 1 - 0.694T + 361T^{2} \)
23 \( 1 + (-23.1 + 23.1i)T - 529iT^{2} \)
29 \( 1 + 49.1T + 841T^{2} \)
31 \( 1 + 33.8iT - 961T^{2} \)
37 \( 1 + (-2.02 + 2.02i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.5T + 1.68e3T^{2} \)
43 \( 1 + (30.4 + 30.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (18.7 - 18.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-33.8 + 33.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 23.1iT - 3.48e3T^{2} \)
61 \( 1 - 12.9iT - 3.72e3T^{2} \)
67 \( 1 + (56.3 - 56.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 92.7iT - 5.04e3T^{2} \)
73 \( 1 + (95.4 - 95.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 100. iT - 6.24e3T^{2} \)
83 \( 1 + (-5.62 - 5.62i)T + 6.88e3iT^{2} \)
89 \( 1 + 158. iT - 7.92e3T^{2} \)
97 \( 1 + (-37.1 - 37.1i)T + 9.40e3iT^{2} \)
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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

−12.74175306027806147065590553577, −11.57977056706389056052912770115, −10.74658218858136199977862832114, −9.819735967698617781136490672752, −8.677641311395851419730042354154, −7.36128328012560576464405924877, −5.90768813814127468623041594032, −4.44975740200054676809720640363, −1.79623594621382108552937919350, −0.37285198986103562626621932439, 3.26415836957764839987433581959, 5.44301377900046536254577196426, 6.45280194335087387941614805239, 7.36255248785204599052831342027, 8.990776600391247484988418422238, 9.668577674534834344189780786135, 10.96504744366471607755964925742, 11.80539364595912122148649003734, 13.10163553925817250628821936237, 14.79419747412341624435027622687

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