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Properties

Label 1-1004-1004.351-r1-0-0
Degree $1$
Conductor $1004$
Sign $0.645 + 0.764i$
Analytic cond. $107.894$
Root an. cond. $107.894$
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  + (0.187 + 0.982i)3-s + (−0.809 − 0.587i)5-s + (−0.535 − 0.844i)7-s + (−0.929 + 0.368i)9-s + (−0.728 − 0.684i)11-s + (−0.929 + 0.368i)13-s + (0.425 − 0.904i)15-s + (0.876 + 0.481i)17-s + (−0.968 + 0.248i)19-s + (0.728 − 0.684i)21-s + (−0.876 − 0.481i)23-s + (0.309 + 0.951i)25-s + (−0.535 − 0.844i)27-s + (−0.637 + 0.770i)29-s + (−0.728 − 0.684i)31-s + ⋯
L(s)  = 1  + (0.187 + 0.982i)3-s + (−0.809 − 0.587i)5-s + (−0.535 − 0.844i)7-s + (−0.929 + 0.368i)9-s + (−0.728 − 0.684i)11-s + (−0.929 + 0.368i)13-s + (0.425 − 0.904i)15-s + (0.876 + 0.481i)17-s + (−0.968 + 0.248i)19-s + (0.728 − 0.684i)21-s + (−0.876 − 0.481i)23-s + (0.309 + 0.951i)25-s + (−0.535 − 0.844i)27-s + (−0.637 + 0.770i)29-s + (−0.728 − 0.684i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1004\)    =    \(2^{2} \cdot 251\)
Sign: $0.645 + 0.764i$
Analytic conductor: \(107.894\)
Root analytic conductor: \(107.894\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1004} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1004,\ (1:\ ),\ 0.645 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3926029313 + 0.1823280717i\)
\(L(\frac12)\) \(\approx\) \(0.3926029313 + 0.1823280717i\)
\(L(1)\) \(\approx\) \(0.6295506395 + 0.05274920440i\)
\(L(1)\) \(\approx\) \(0.6295506395 + 0.05274920440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 \)
good3 \( 1 + (0.187 + 0.982i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.535 - 0.844i)T \)
11 \( 1 + (-0.728 - 0.684i)T \)
13 \( 1 + (-0.929 + 0.368i)T \)
17 \( 1 + (0.876 + 0.481i)T \)
19 \( 1 + (-0.968 + 0.248i)T \)
23 \( 1 + (-0.876 - 0.481i)T \)
29 \( 1 + (-0.637 + 0.770i)T \)
31 \( 1 + (-0.728 - 0.684i)T \)
37 \( 1 + (-0.929 + 0.368i)T \)
41 \( 1 + (-0.425 - 0.904i)T \)
43 \( 1 + (-0.0627 - 0.998i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.0627 - 0.998i)T \)
59 \( 1 + (-0.876 + 0.481i)T \)
61 \( 1 + (0.0627 - 0.998i)T \)
67 \( 1 + (0.425 + 0.904i)T \)
71 \( 1 + (0.929 - 0.368i)T \)
73 \( 1 + (-0.187 + 0.982i)T \)
79 \( 1 + (0.637 + 0.770i)T \)
83 \( 1 + (0.992 - 0.125i)T \)
89 \( 1 + (0.876 + 0.481i)T \)
97 \( 1 + (-0.992 + 0.125i)T \)
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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

−21.394183615839397982558414498895, −20.29746681323387311114176870416, −19.55119408537167498080809682428, −19.06550296436957441912372774645, −18.29721695364800499093855193943, −17.75754331213511976205026113133, −16.644331666646789327568420169465, −15.62965837789655760425960688560, −14.995951143732727376055405854683, −14.35631719951440914689618135773, −13.22967835613542242968760659261, −12.35897713736153600869777092299, −12.131770443766338723968082940292, −11.0773435206627683843588916700, −10.032544733231976020806999290982, −9.15405552578981462936797739567, −7.95076431127222363659506772804, −7.6270025992458361459139032976, −6.69926811621993165346090547306, −5.83837885030126826708242178127, −4.82780587878818923552971593834, −3.40812904860327055683791727105, −2.70452971366118337785167196767, −1.91307079268862265067824498623, −0.21239209871124539191140172263, 0.3795606879007001023508247093, 2.11481932657063162075583396652, 3.53640335664334326770100131759, 3.810158035575340806080234065350, 4.886938434014443223939883205145, 5.62573104664218751273529546081, 6.93971669463601295175773807988, 7.94024196454471619022080489423, 8.527278026972522622978633569329, 9.53494927923298560891229908430, 10.35699464498276959847686142306, 10.89983733751892255520859248260, 12.004968320878205964796862607523, 12.74553616799336478526741653307, 13.73182965333578468487470257385, 14.57780969200482672218075702830, 15.32410173226235577704590345280, 16.21631073437615471748777997541, 16.681350217136580592428150712313, 17.1658291385724085557586730256, 18.78953865922560903113699967977, 19.30508754002597258964976605744, 20.181103853482815864229058142858, 20.628376492032439150487947666111, 21.53468793804909904759818903043

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