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Properties

Label 1-2624-2624.1373-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.199 + 0.979i$
Analytic cond. $12.1858$
Root an. cond. $12.1858$
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  + (0.382 − 0.923i)3-s + (−0.760 + 0.649i)5-s + (0.987 + 0.156i)7-s + (−0.707 − 0.707i)9-s + (0.996 − 0.0784i)11-s + (−0.522 − 0.852i)13-s + (0.309 + 0.951i)15-s + (−0.309 + 0.951i)17-s + (−0.972 + 0.233i)19-s + (0.522 − 0.852i)21-s + (0.156 + 0.987i)23-s + (0.156 − 0.987i)25-s + (−0.923 + 0.382i)27-s + (−0.996 − 0.0784i)29-s + (−0.309 + 0.951i)31-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.760 + 0.649i)5-s + (0.987 + 0.156i)7-s + (−0.707 − 0.707i)9-s + (0.996 − 0.0784i)11-s + (−0.522 − 0.852i)13-s + (0.309 + 0.951i)15-s + (−0.309 + 0.951i)17-s + (−0.972 + 0.233i)19-s + (0.522 − 0.852i)21-s + (0.156 + 0.987i)23-s + (0.156 − 0.987i)25-s + (−0.923 + 0.382i)27-s + (−0.996 − 0.0784i)29-s + (−0.309 + 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.199 + 0.979i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.199 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8024913219 + 0.6557696064i\)
\(L(\frac12)\) \(\approx\) \(0.8024913219 + 0.6557696064i\)
\(L(1)\) \(\approx\) \(1.000479455 - 0.05609402199i\)
\(L(1)\) \(\approx\) \(1.000479455 - 0.05609402199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (-0.760 + 0.649i)T \)
7 \( 1 + (0.987 + 0.156i)T \)
11 \( 1 + (0.996 - 0.0784i)T \)
13 \( 1 + (-0.522 - 0.852i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.972 + 0.233i)T \)
23 \( 1 + (0.156 + 0.987i)T \)
29 \( 1 + (-0.996 - 0.0784i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.760 + 0.649i)T \)
43 \( 1 + (0.972 + 0.233i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.760 + 0.649i)T \)
59 \( 1 + (-0.233 + 0.972i)T \)
61 \( 1 + (0.972 - 0.233i)T \)
67 \( 1 + (-0.996 - 0.0784i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (-0.987 - 0.156i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−19.30983670845151257446860744962, −18.74056097050293188038823933359, −17.44991385143910447059902113294, −16.960463581225751361306462838797, −16.40968666497111374869478688685, −15.6490587702032946366105989984, −14.784125446002390185372534821804, −14.51723988480071590634388805001, −13.70293755126233798011229371121, −12.68286773512660210240300174357, −11.78324560351195690776824973912, −11.328655374376591112658317356292, −10.686456357465236322042464556991, −9.55245337331929575038007385198, −8.96547229423996742612990981182, −8.53105466724046027099849640825, −7.56823417385013641904186140861, −6.918830419048137072871269789156, −5.61508503102524524614777254656, −4.766608666793021237782438377747, −4.271601199209944777923761078565, −3.81354889341122616027874980312, −2.486823313325347772982185143280, −1.720483187057534419998324655845, −0.30792491372273408853002180322, 1.222261136054756936198298296484, 1.90885847902083764501370851233, 2.88646463239814879070477128674, 3.68692616288304266976796922021, 4.45219344896459425128326869316, 5.683638043337030178846742292505, 6.32973794911496626964598380374, 7.3549266478739332608830671805, 7.61388744644822139750105070400, 8.54954899743797603767259358844, 8.98010542408646764853396386265, 10.32178872687155155072696336179, 11.0210789240754363975772151278, 11.692153253958477755218246555044, 12.31228194329271574616491325690, 12.97577496798599153878459716151, 13.97960442219844786725894836116, 14.65631948814016447731646084854, 14.94393700544660662224336888670, 15.69042893873498518150602616183, 17.10453075475998942186793612814, 17.40015731071541883782684981835, 18.09876077560683137276230448501, 19.02908854864973287782198474469, 19.32819683653147785610758558130

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