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Properties

Label 1-2015-2015.997-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.115 + 0.993i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + i·17-s + (0.866 + 0.5i)18-s − 19-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + i·17-s + (0.866 + 0.5i)18-s − 19-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.115 + 0.993i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.115 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.619013846 + 3.222574188i\)
\(L(\frac12)\) \(\approx\) \(3.619013846 + 3.222574188i\)
\(L(1)\) \(\approx\) \(2.270591680 + 0.2241177133i\)
\(L(1)\) \(\approx\) \(2.270591680 + 0.2241177133i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
17 \( 1 + iT \)
19 \( 1 - T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + iT \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.85665679509379349754466636011, −19.04819577264775711815957179702, −17.97452029354467373502165397134, −17.31642631432266978927836679264, −16.57862359532328027078851652496, −15.83388238095941193742479645513, −14.92049341511267129875947869628, −14.36455062793833626053164820699, −13.75737962890813789365147388202, −13.30577951698762296240716786846, −12.3326501110981476641978081031, −11.82231509455183645403857623021, −10.80165639038422465241587375221, −9.78898334819538433192842312844, −8.89106889230834739797013673554, −8.06868068810395895403475355860, −7.46653525734513438978820115260, −6.58570886806634827030116010187, −6.29618683472354591667596523005, −4.79886867556009452429621782860, −4.123211795536557006719525201844, −3.50581236422116181171173019811, −2.53911627941344587484076534275, −1.664066325131511205494648735463, −0.45781587877155576127276617324, 1.40720591968454909511647835959, 2.07916538301132556669367838226, 2.870815233222578101689710394206, 3.8307066331491243248352458463, 4.25282300240341237790711992338, 5.32762147568706872762623496337, 6.06024467517244874661054467590, 6.901270305986731094435121239372, 8.06934644833567961160886256361, 8.9316405656752328412807945376, 9.41999266753525630728004008792, 10.43672957938257097990298234094, 10.94754124009295824841945207390, 12.114410487695420200554563808393, 12.44522158626851016128375148435, 13.395690338827273269394938973377, 14.19927469603873440091329533428, 14.696977137103252221521313528181, 15.32339640187438142105132840391, 15.91220366388639965256765812658, 16.79821907706771464383809842241, 17.90784045604180195001960216073, 18.89332325951818743598996209002, 19.51517328319960597383389806557, 19.78752721218780361741341309189

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