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Properties

Label 1-2103-2103.1217-r0-0-0
Degree $1$
Conductor $2103$
Sign $-0.992 - 0.118i$
Analytic cond. $9.76629$
Root an. cond. $9.76629$
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  + (0.125 − 0.992i)2-s + (−0.968 − 0.248i)4-s + (0.0627 − 0.998i)5-s + (−0.929 + 0.368i)7-s + (−0.368 + 0.929i)8-s + (−0.982 − 0.187i)10-s + (−0.248 − 0.968i)11-s + (0.0627 + 0.998i)13-s + (0.248 + 0.968i)14-s + (0.876 + 0.481i)16-s + (0.309 + 0.951i)17-s + 19-s + (−0.309 + 0.951i)20-s + (−0.992 + 0.125i)22-s + (0.587 − 0.809i)23-s + ⋯
L(s)  = 1  + (0.125 − 0.992i)2-s + (−0.968 − 0.248i)4-s + (0.0627 − 0.998i)5-s + (−0.929 + 0.368i)7-s + (−0.368 + 0.929i)8-s + (−0.982 − 0.187i)10-s + (−0.248 − 0.968i)11-s + (0.0627 + 0.998i)13-s + (0.248 + 0.968i)14-s + (0.876 + 0.481i)16-s + (0.309 + 0.951i)17-s + 19-s + (−0.309 + 0.951i)20-s + (−0.992 + 0.125i)22-s + (0.587 − 0.809i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2103\)    =    \(3 \cdot 701\)
Sign: $-0.992 - 0.118i$
Analytic conductor: \(9.76629\)
Root analytic conductor: \(9.76629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2103} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2103,\ (0:\ ),\ -0.992 - 0.118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06285772481 - 1.055390134i\)
\(L(\frac12)\) \(\approx\) \(0.06285772481 - 1.055390134i\)
\(L(1)\) \(\approx\) \(0.6744677852 - 0.5965962741i\)
\(L(1)\) \(\approx\) \(0.6744677852 - 0.5965962741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
701 \( 1 \)
good2 \( 1 + (0.125 - 0.992i)T \)
5 \( 1 + (0.0627 - 0.998i)T \)
7 \( 1 + (-0.929 + 0.368i)T \)
11 \( 1 + (-0.248 - 0.968i)T \)
13 \( 1 + (0.0627 + 0.998i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.728 - 0.684i)T \)
31 \( 1 + (0.992 + 0.125i)T \)
37 \( 1 + (-0.125 + 0.992i)T \)
41 \( 1 + (0.535 - 0.844i)T \)
43 \( 1 + (-0.535 - 0.844i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.684 + 0.728i)T \)
59 \( 1 + (-0.904 - 0.425i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (0.998 - 0.0627i)T \)
71 \( 1 + (-0.684 - 0.728i)T \)
73 \( 1 + (-0.844 - 0.535i)T \)
79 \( 1 + (0.684 - 0.728i)T \)
83 \( 1 + (-0.968 + 0.248i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.968 - 0.248i)T \)
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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

−20.06667465418480209178303076322, −19.343521751664178933812792137468, −18.429236522675770788697275311250, −17.9970675412080957109502482427, −17.35203391517955232906459786535, −16.416042360891584473800634874321, −15.72094928109987473280870764381, −15.22776226842547385352320849679, −14.46912991471533297373398125293, −13.70930558849194334791918129872, −13.10988380032924811621886042198, −12.37608663526684714646395536790, −11.35591032187731846508504443677, −10.2619969284300804031595906691, −9.788299275354917904769769303171, −9.16482110498665319083632928333, −7.79332417528823494390078232317, −7.34588460929429926700504928604, −6.85088859500871890314494478465, −5.85330922060593024684497930456, −5.267800200362820003312202003762, −4.18049392831399144772421918269, −3.22888834944941734324237900261, −2.77613825295186714227638858864, −1.00988642025097805905384488296, 0.4275664117553819891292724784, 1.37672406087894412631016616493, 2.36543170074448973295821322866, 3.30066376000552865716275255465, 3.99586875413815287441200375142, 4.90641785145009284720018214646, 5.72402209910485724820687403156, 6.34811541929596124766648679897, 7.76503935760017183059185923914, 8.74592680099215792163532942283, 9.022120636687847439137541622299, 9.8873229550856164593942246374, 10.58833457853144065192370554445, 11.61804947012300557065711302317, 12.13159261106397518559722146854, 12.79196746720401800599719644010, 13.59448294204882840736601527095, 13.90454927257967768746260895540, 15.15900857971185901871820518186, 15.91998638690851576710995868135, 16.83182803939544851281623985236, 17.11706812279594737797430356639, 18.45972199570221420868613303453, 18.92819256125266638346448235306, 19.41479690247700291795388035968

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