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Properties

Label 1-1968-1968.251-r0-0-0
Degree $1$
Conductor $1968$
Sign $0.00791 - 0.999i$
Analytic cond. $9.13935$
Root an. cond. $9.13935$
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  + (0.587 − 0.809i)5-s + (−0.951 + 0.309i)7-s + (0.809 − 0.587i)11-s + (0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.309 + 0.951i)35-s + (0.587 − 0.809i)37-s + (0.951 + 0.309i)43-s + (−0.951 − 0.309i)47-s + (0.809 − 0.587i)49-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)5-s + (−0.951 + 0.309i)7-s + (0.809 − 0.587i)11-s + (0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.309 + 0.951i)35-s + (0.587 − 0.809i)37-s + (0.951 + 0.309i)43-s + (−0.951 − 0.309i)47-s + (0.809 − 0.587i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00791 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00791 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1968\)    =    \(2^{4} \cdot 3 \cdot 41\)
Sign: $0.00791 - 0.999i$
Analytic conductor: \(9.13935\)
Root analytic conductor: \(9.13935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1968} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1968,\ (0:\ ),\ 0.00791 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.097465479 - 1.088809263i\)
\(L(\frac12)\) \(\approx\) \(1.097465479 - 1.088809263i\)
\(L(1)\) \(\approx\) \(1.068508008 - 0.3055788631i\)
\(L(1)\) \(\approx\) \(1.068508008 - 0.3055788631i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 - iT \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−19.96477023964506566959330610845, −19.439847189726545284538402881492, −18.812614271012735950509301046060, −17.902432585855545184309727084, −17.31858487372714547400350726787, −16.625529011941004140457530807364, −15.72033343811707190160740336833, −15.07339087702214863674808396718, −14.12373751962225651883688352369, −13.73373510567194151121714523777, −12.86344562978175899649668539179, −12.06285471096386396619395669120, −11.1702835126171647600207425524, −10.46675573516899938311035053061, −9.66760695015268173388976308992, −9.20222298683823403768501124861, −8.170380617727527521744647819913, −6.87881107760581984309548774758, −6.63469209860308657261399222208, −6.06303363821660907579727743457, −4.639073532807472387616696519444, −4.00207831809145654078332361338, −2.97283311525784345214153340539, −2.241828595832620276874929459383, −1.17439345686956800140932366710, 0.57958352731184398347218165750, 1.5091571175279980870069250936, 2.68479686142784568330003276087, 3.46388814746632292174106869531, 4.40396197098121270513873410698, 5.5091555576200227237010633317, 5.95901430715266433585891447053, 6.73489006165753966043893974351, 7.89845701402452539762980964285, 8.63486684259417693534545930505, 9.46041523451038343412099138494, 9.81924794508188281072722331124, 10.868778911157089351564939007760, 11.85117351940084439146183886657, 12.43707383219093292104849275223, 13.266372210967373098398339880773, 13.7248418989678080244863473158, 14.595575641481880845592238590594, 15.72795656840179689080955178111, 16.123658888828689897825188610711, 16.79107179528009908065654393619, 17.70115407112282862688182230726, 18.184784076546551783844379425734, 19.269473701963454650379328072406, 19.78162704285496920115198665496

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