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Properties

Label 2-1520-1520.949-c0-0-5
Degree $2$
Conductor $1520$
Sign $0.923 - 0.382i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (−0.980 + 0.195i)2-s + (0.275 − 0.275i)3-s + (0.923 − 0.382i)4-s + (0.707 + 0.707i)5-s + (−0.216 + 0.324i)6-s + (−0.831 + 0.555i)8-s + 0.847i·9-s + (−0.831 − 0.555i)10-s + (−0.541 − 0.541i)11-s + (0.149 − 0.360i)12-s + (0.785 − 0.785i)13-s + 0.390·15-s + (0.707 − 0.707i)16-s + (−0.165 − 0.831i)18-s + (0.707 − 0.707i)19-s + (0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)2-s + (0.275 − 0.275i)3-s + (0.923 − 0.382i)4-s + (0.707 + 0.707i)5-s + (−0.216 + 0.324i)6-s + (−0.831 + 0.555i)8-s + 0.847i·9-s + (−0.831 − 0.555i)10-s + (−0.541 − 0.541i)11-s + (0.149 − 0.360i)12-s + (0.785 − 0.785i)13-s + 0.390·15-s + (0.707 − 0.707i)16-s + (−0.165 − 0.831i)18-s + (0.707 − 0.707i)19-s + (0.923 + 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9199584875\)
\(L(\frac12)\) \(\approx\) \(0.9199584875\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 - 0.195i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
13 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
67 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.11T + T^{2} \)
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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

−9.790019926893020149310404019874, −8.812420463493615495944967425649, −8.085818328998989269941273502370, −7.51099965625427066454983061457, −6.60356660641400198848460727457, −5.84964008968626169034411544352, −5.07572483897199131807252905547, −3.14319568199063472739406522516, −2.59259490689797777190493511251, −1.35704140994080236832109299407, 1.19833927235473156341067808003, 2.25603028680066625741206286979, 3.44866343149143682165035361677, 4.44825387932936429858409441047, 5.76408114663102189835350435546, 6.38131073267588228343635247619, 7.41432917541547060339547803458, 8.256970750859021669266161021550, 9.032387428836150261172541093916, 9.509779784053368896543290155404

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