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Properties

Label 1-2404-2404.1103-r0-0-0
Degree $1$
Conductor $2404$
Sign $-0.993 + 0.109i$
Analytic cond. $11.1641$
Root an. cond. $11.1641$
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  + (0.809 + 0.587i)3-s + i·5-s + (−0.156 + 0.987i)7-s + (0.309 + 0.951i)9-s + (0.453 + 0.891i)11-s + i·13-s + (−0.587 + 0.809i)15-s + (0.707 − 0.707i)17-s + (−0.156 + 0.987i)19-s + (−0.707 + 0.707i)21-s + (−0.951 − 0.309i)23-s − 25-s + (−0.309 + 0.951i)27-s + (0.987 + 0.156i)29-s + (−0.891 + 0.453i)31-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + i·5-s + (−0.156 + 0.987i)7-s + (0.309 + 0.951i)9-s + (0.453 + 0.891i)11-s + i·13-s + (−0.587 + 0.809i)15-s + (0.707 − 0.707i)17-s + (−0.156 + 0.987i)19-s + (−0.707 + 0.707i)21-s + (−0.951 − 0.309i)23-s − 25-s + (−0.309 + 0.951i)27-s + (0.987 + 0.156i)29-s + (−0.891 + 0.453i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2404 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2404\)    =    \(2^{2} \cdot 601\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(11.1641\)
Root analytic conductor: \(11.1641\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2404} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2404,\ (0:\ ),\ -0.993 + 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1161673127 + 2.109497738i\)
\(L(\frac12)\) \(\approx\) \(0.1161673127 + 2.109497738i\)
\(L(1)\) \(\approx\) \(1.041110313 + 0.9103854118i\)
\(L(1)\) \(\approx\) \(1.041110313 + 0.9103854118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
601 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + iT \)
7 \( 1 + (-0.156 + 0.987i)T \)
11 \( 1 + (0.453 + 0.891i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (-0.891 + 0.453i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (0.891 - 0.453i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.453 - 0.891i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.987 - 0.156i)T \)
79 \( 1 + (0.987 + 0.156i)T \)
83 \( 1 + (0.891 + 0.453i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−19.57687011265851726997598618937, −18.5830157368157991685286899405, −17.63871498039885103600415201564, −17.17000390478067971704045055694, −16.3643780241172064687712901000, −15.639705896863760701247034385812, −14.80800942447722930418261689341, −13.85526733045848819091009132946, −13.5750544722385952544479280717, −12.76463928686753609236112006942, −12.25541127512438165180070753377, −11.26286791936437494788585386145, −10.312323362403046988915835165684, −9.57624632182073309995932144415, −8.78421205074124786734395458663, −8.07875861584905325439216092407, −7.65982260650801763200146072718, −6.58680165511880888480900632152, −5.88314304464343895628193287995, −4.863715178320659101682033216757, −3.79772157512169099757778968609, −3.43614610254601009813831165763, −2.23087629255129401116819600342, −1.15663587880484763038705166736, −0.64035636300963632456117691888, 1.892520159059595291763176671884, 2.19416974865418533465437516866, 3.26877225806762968284728840696, 3.84270248345706517280473294331, 4.79422234482360599132430352955, 5.71195989238551951652496498838, 6.66000421535075505081477780499, 7.32865675661883130816674622705, 8.20393234490371286616301226459, 9.01389039381003334832499842247, 9.70772691935494307954391737526, 10.185267562882062501835332160807, 11.0930668905099235137348678657, 12.070525616895973626298984437, 12.43275111198351687585602749284, 13.835261261771314132416112313877, 14.350230178643671110335647648605, 14.63943524275276191224019788980, 15.651386207458018903547611740156, 15.99140080763541040494115680351, 16.927032983533633787201460640225, 18.02017503952612319694884317106, 18.53374560784220440466750432999, 19.23406231406624508723852228158, 19.69455046639518383305894169744

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