Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.790 - 0.612i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.28 − 2.28i)2-s + (−1.07 − 2.80i)3-s + 6.40i·4-s + (1.80 + 4.66i)5-s + (−3.93 + 8.83i)6-s + (−6.22 + 3.20i)7-s + (5.48 − 5.48i)8-s + (−6.69 + 6.01i)9-s + (6.50 − 14.7i)10-s + 11.1i·11-s + (17.9 − 6.88i)12-s + (5.82 − 5.82i)13-s + (21.5 + 6.86i)14-s + (11.1 − 10.0i)15-s + 0.592·16-s + (6.84 − 6.84i)17-s + ⋯
L(s)  = 1  + (−1.14 − 1.14i)2-s + (−0.358 − 0.933i)3-s + 1.60i·4-s + (0.361 + 0.932i)5-s + (−0.656 + 1.47i)6-s + (−0.888 + 0.458i)7-s + (0.685 − 0.685i)8-s + (−0.743 + 0.668i)9-s + (0.650 − 1.47i)10-s + 1.01i·11-s + (1.49 − 0.573i)12-s + (0.448 − 0.448i)13-s + (1.53 + 0.490i)14-s + (0.740 − 0.671i)15-s + 0.0370·16-s + (0.402 − 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.790 - 0.612i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.790 - 0.612i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.316757 + 0.108374i\)
\(L(\frac12)\)  \(\approx\)  \(0.316757 + 0.108374i\)
\(L(2)\)   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 \( 1 + (1.07 + 2.80i)T \)
5 \( 1 + (-1.80 - 4.66i)T \)
7 \( 1 + (6.22 - 3.20i)T \)
good2 \( 1 + (2.28 + 2.28i)T + 4iT^{2} \)
11 \( 1 - 11.1iT - 121T^{2} \)
13 \( 1 + (-5.82 + 5.82i)T - 169iT^{2} \)
17 \( 1 + (-6.84 + 6.84i)T - 289iT^{2} \)
19 \( 1 + 25.0T + 361T^{2} \)
23 \( 1 + (23.3 - 23.3i)T - 529iT^{2} \)
29 \( 1 - 10.6T + 841T^{2} \)
31 \( 1 - 26.9iT - 961T^{2} \)
37 \( 1 + (20.8 - 20.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.9T + 1.68e3T^{2} \)
43 \( 1 + (-1.25 - 1.25i)T + 1.84e3iT^{2} \)
47 \( 1 + (59.1 - 59.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-26.0 + 26.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 70.2iT - 3.48e3T^{2} \)
61 \( 1 - 14.1iT - 3.72e3T^{2} \)
67 \( 1 + (6.14 - 6.14i)T - 4.48e3iT^{2} \)
71 \( 1 - 39.0iT - 5.04e3T^{2} \)
73 \( 1 + (51.1 - 51.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 16.8iT - 6.24e3T^{2} \)
83 \( 1 + (31.3 + 31.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 70.8iT - 7.92e3T^{2} \)
97 \( 1 + (114. + 114. i)T + 9.40e3iT^{2} \)
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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

−13.10487080764373714713260924464, −12.36212661154777173417101184993, −11.43702603348641561641457297095, −10.39982095847710024432405318213, −9.659146273287731680338360465122, −8.313212414828118838160900083060, −7.10681961836098273881963238698, −5.96180207190513305514522380492, −3.07777932648089067216142118021, −1.89711609327720206838822880085, 0.36324139024709615945713561161, 4.05965943790312508621181356171, 5.80765555511973259586416325511, 6.41792335236287531883983700983, 8.293067537942761375803907766796, 8.946134196615351254338342898426, 9.919281738064346247157895869514, 10.70175417218389912798897786794, 12.34501149345211348937499658347, 13.68735122459342608897275223050

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