Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.999 + 0.0338i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (2.70 + 1.29i)3-s + 3i·4-s + (0.707 − 4.94i)5-s + (2.82 − i)6-s − 7i·7-s + (4.94 + 4.94i)8-s + (5.65 + 7i)9-s + (−3.00 − 4.00i)10-s + 9.89i·11-s + (−3.87 + 8.12i)12-s + (8 + 8i)13-s + (−4.94 − 4.94i)14-s + (8.31 − 12.4i)15-s − 4.99·16-s + (−18.3 − 18.3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.353i)2-s + (0.902 + 0.430i)3-s + 0.750i·4-s + (0.141 − 0.989i)5-s + (0.471 − 0.166i)6-s i·7-s + (0.618 + 0.618i)8-s + (0.628 + 0.777i)9-s + (−0.300 − 0.400i)10-s + 0.899i·11-s + (−0.323 + 0.676i)12-s + (0.615 + 0.615i)13-s + (−0.353 − 0.353i)14-s + (0.554 − 0.832i)15-s − 0.312·16-s + (−1.08 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.999 + 0.0338i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.999 + 0.0338i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.09602 - 0.0355141i\)
\(L(\frac12)\)  \(\approx\)  \(2.09602 - 0.0355141i\)
\(L(2)\)   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 \( 1 + (-2.70 - 1.29i)T \)
5 \( 1 + (-0.707 + 4.94i)T \)
7 \( 1 + 7iT \)
good2 \( 1 + (-0.707 + 0.707i)T - 4iT^{2} \)
11 \( 1 - 9.89iT - 121T^{2} \)
13 \( 1 + (-8 - 8i)T + 169iT^{2} \)
17 \( 1 + (18.3 + 18.3i)T + 289iT^{2} \)
19 \( 1 + 10T + 361T^{2} \)
23 \( 1 + (24.0 + 24.0i)T + 529iT^{2} \)
29 \( 1 + 4.24T + 841T^{2} \)
31 \( 1 + 14iT - 961T^{2} \)
37 \( 1 + (-30 - 30i)T + 1.36e3iT^{2} \)
41 \( 1 + 33.9T + 1.68e3T^{2} \)
43 \( 1 + (-36 + 36i)T - 1.84e3iT^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 2.20e3iT^{2} \)
53 \( 1 + (-70.7 - 70.7i)T + 2.80e3iT^{2} \)
59 \( 1 + 29.6iT - 3.48e3T^{2} \)
61 \( 1 - 14iT - 3.72e3T^{2} \)
67 \( 1 + (-32 - 32i)T + 4.48e3iT^{2} \)
71 \( 1 - 59.3iT - 5.04e3T^{2} \)
73 \( 1 + (-39 - 39i)T + 5.32e3iT^{2} \)
79 \( 1 + 56iT - 6.24e3T^{2} \)
83 \( 1 + (36.7 - 36.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 19.7iT - 7.92e3T^{2} \)
97 \( 1 + (-113 + 113i)T - 9.40e3iT^{2} \)
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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

−13.48925061161468557548554797153, −12.75942651533726075640658333542, −11.55104778079896122386114604078, −10.25481035052870926872481931563, −9.091765553966118946111961630164, −8.193493020369552392437613479497, −7.05945642683123586463011996937, −4.54679444261208009815435983861, −4.11642442026930395648732055817, −2.19475981152316429990697946482, 2.08613333671172545868103919682, 3.68280546313233073517983183157, 5.85132540180852832653601515936, 6.48996326215910730911536666978, 7.990704370693596228110361884251, 9.089146681742194004043493140220, 10.31566467208221908516233569754, 11.33982009597783489700585662681, 12.94314844795436462932885353596, 13.69018909164041274046321438987

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