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Properties

Label 2-3520-220.87-c0-0-5
Degree $2$
Conductor $3520$
Sign $-0.999 + 0.0299i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 0.366i)3-s + (−0.866 − 0.5i)5-s − 0.732i·9-s − i·11-s + (0.133 + 0.5i)15-s + (−1.36 − 1.36i)23-s + (0.499 + 0.866i)25-s + (−0.633 + 0.633i)27-s + i·31-s + (−0.366 + 0.366i)33-s + (0.366 + 0.366i)37-s + (−0.366 + 0.633i)45-s + (−1 + i)47-s − i·49-s + (1 − i)53-s + ⋯
L(s)  = 1  + (−0.366 − 0.366i)3-s + (−0.866 − 0.5i)5-s − 0.732i·9-s − i·11-s + (0.133 + 0.5i)15-s + (−1.36 − 1.36i)23-s + (0.499 + 0.866i)25-s + (−0.633 + 0.633i)27-s + i·31-s + (−0.366 + 0.366i)33-s + (0.366 + 0.366i)37-s + (−0.366 + 0.633i)45-s + (−1 + i)47-s − i·49-s + (1 − i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.999 + 0.0299i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.999 + 0.0299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4655333513\)
\(L(\frac12)\) \(\approx\) \(0.4655333513\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + iT \)
good3 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.404878217187923918154263527227, −7.74410611981644074808021501215, −6.79452288205893017782299993568, −6.24238451158614976905134171180, −5.43085957637743082364506314566, −4.48559783970044259250182539723, −3.73570617968852139911962425051, −2.91720505887471695606093219709, −1.42402016148136689242842073260, −0.29142745403466414342809404258, 1.79192709882737053832664303168, 2.77477209896021802487901134215, 3.95335025811482280535672987251, 4.36300983500307617052202744467, 5.32052859328401642483734147173, 6.07535275547887912679904707010, 7.04786165208031453519921698090, 7.72748371545842967980730193285, 8.057162344082686815627454513117, 9.261050291883630907317447557235

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