Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.824 + 0.565i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.992 + 0.992i)2-s + (1.22 − 1.22i)3-s − 2.02i·4-s + (−2.01 − 4.57i)5-s + 2.43·6-s + (1.87 + 1.87i)7-s + (5.98 − 5.98i)8-s − 2.99i·9-s + (2.53 − 6.54i)10-s + 6.89·11-s + (−2.48 − 2.48i)12-s + (−11.8 + 11.8i)13-s + 3.71i·14-s + (−8.07 − 3.12i)15-s + 3.77·16-s + (16.7 + 16.7i)17-s + ⋯
L(s)  = 1  + (0.496 + 0.496i)2-s + (0.408 − 0.408i)3-s − 0.507i·4-s + (−0.403 − 0.914i)5-s + 0.405·6-s + (0.267 + 0.267i)7-s + (0.748 − 0.748i)8-s − 0.333i·9-s + (0.253 − 0.654i)10-s + 0.627·11-s + (−0.206 − 0.206i)12-s + (−0.914 + 0.914i)13-s + 0.265i·14-s + (−0.538 − 0.208i)15-s + 0.235·16-s + (0.988 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.824 + 0.565i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (22, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.824 + 0.565i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.79816 - 0.557654i\)
\(L(\frac12)\)  \(\approx\)  \(1.79816 - 0.557654i\)
\(L(2)\)   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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (2.01 + 4.57i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (-0.992 - 0.992i)T + 4iT^{2} \)
11 \( 1 - 6.89T + 121T^{2} \)
13 \( 1 + (11.8 - 11.8i)T - 169iT^{2} \)
17 \( 1 + (-16.7 - 16.7i)T + 289iT^{2} \)
19 \( 1 - 8.54iT - 361T^{2} \)
23 \( 1 + (-12.4 + 12.4i)T - 529iT^{2} \)
29 \( 1 + 1.33iT - 841T^{2} \)
31 \( 1 + 18.4T + 961T^{2} \)
37 \( 1 + (-31.4 - 31.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.7T + 1.68e3T^{2} \)
43 \( 1 + (15.5 - 15.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (22.1 + 22.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-66.4 + 66.4i)T - 2.80e3iT^{2} \)
59 \( 1 - 81.8iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + (-79.2 - 79.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 63.1T + 5.04e3T^{2} \)
73 \( 1 + (-92.9 + 92.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 8.46iT - 6.24e3T^{2} \)
83 \( 1 + (-36.2 + 36.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (79.2 + 79.2i)T + 9.40e3iT^{2} \)
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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

−13.53183030767607036865962290087, −12.53020487633220336409377430330, −11.68622871617937809924862910568, −10.02592636452300679879211990013, −8.958151367907325216106003066085, −7.79191318348194601908388926002, −6.56874286181063221074159265541, −5.25176845531168871191566466448, −4.06970388482442401544799881781, −1.49957020755019577682609859025, 2.72933658145651962435059370241, 3.70086298943075642902753378801, 5.07949952756173782805355322392, 7.20931057829597836231638168481, 7.913774453453327156338830005901, 9.489235453808949592318744139297, 10.68789435171295379887166729683, 11.53967546102041911831916885979, 12.48282670865517362745352481584, 13.74904664268695811497687596694

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