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Properties

Label 1-71e2-5041.18-r0-0-0
Degree $1$
Conductor $5041$
Sign $-0.358 - 0.933i$
Analytic cond. $23.4103$
Root an. cond. $23.4103$
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  + (−0.882 + 0.469i)2-s + (0.0259 − 0.999i)3-s + (0.558 − 0.829i)4-s + (0.994 + 0.105i)5-s + (0.446 + 0.894i)6-s + (0.226 − 0.974i)7-s + (−0.102 + 0.994i)8-s + (−0.998 − 0.0518i)9-s + (−0.927 + 0.373i)10-s + (−0.393 − 0.919i)11-s + (−0.814 − 0.579i)12-s + (0.654 − 0.755i)13-s + (0.258 + 0.966i)14-s + (0.131 − 0.991i)15-s + (−0.376 − 0.926i)16-s + (−0.416 + 0.909i)17-s + ⋯
L(s)  = 1  + (−0.882 + 0.469i)2-s + (0.0259 − 0.999i)3-s + (0.558 − 0.829i)4-s + (0.994 + 0.105i)5-s + (0.446 + 0.894i)6-s + (0.226 − 0.974i)7-s + (−0.102 + 0.994i)8-s + (−0.998 − 0.0518i)9-s + (−0.927 + 0.373i)10-s + (−0.393 − 0.919i)11-s + (−0.814 − 0.579i)12-s + (0.654 − 0.755i)13-s + (0.258 + 0.966i)14-s + (0.131 − 0.991i)15-s + (−0.376 − 0.926i)16-s + (−0.416 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ -0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8579254818 - 1.247965982i\)
\(L(\frac12)\) \(\approx\) \(0.8579254818 - 1.247965982i\)
\(L(1)\) \(\approx\) \(0.8471028745 - 0.3633123005i\)
\(L(1)\) \(\approx\) \(0.8471028745 - 0.3633123005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.882 + 0.469i)T \)
3 \( 1 + (0.0259 - 0.999i)T \)
5 \( 1 + (0.994 + 0.105i)T \)
7 \( 1 + (0.226 - 0.974i)T \)
11 \( 1 + (-0.393 - 0.919i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (-0.416 + 0.909i)T \)
19 \( 1 + (-0.468 + 0.883i)T \)
23 \( 1 + (0.191 - 0.981i)T \)
29 \( 1 + (0.877 - 0.479i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.886 - 0.462i)T \)
41 \( 1 + (-0.503 + 0.863i)T \)
43 \( 1 + (0.912 + 0.409i)T \)
47 \( 1 + (-0.329 - 0.944i)T \)
53 \( 1 + (0.491 - 0.870i)T \)
59 \( 1 + (0.952 - 0.306i)T \)
61 \( 1 + (0.644 - 0.764i)T \)
67 \( 1 + (0.811 - 0.584i)T \)
73 \( 1 + (-0.364 + 0.931i)T \)
79 \( 1 + (0.0738 + 0.997i)T \)
83 \( 1 + (0.0233 + 0.999i)T \)
89 \( 1 + (0.983 + 0.181i)T \)
97 \( 1 + (0.958 + 0.286i)T \)
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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

−17.95818671420388520805522274219, −17.71987901492817678824667404113, −17.13365951048263764986538558810, −16.161064383352785550324929832902, −15.77970725394356536267875361781, −15.196263177919183898654796598329, −14.30182658706557155031940370754, −13.52252055111941449322616817421, −12.82201461836228588339682080977, −11.8547687214398488114928768215, −11.48743557808579453951895905452, −10.61361868523025648156098509174, −10.10717257539465745085463894876, −9.32970890145345082085997931409, −8.98461612960682843587354144249, −8.539531313901810719429120603219, −7.39577020025805761381679746453, −6.57720378813713981627033966803, −5.85871049623585994490404170184, −4.89379481127005603807291972196, −4.48017106580397632919381369990, −3.24637182711919006427880453784, −2.517930720801171418018865802569, −2.11164582589107460965668836519, −1.03890829342380978897904141678, 0.65530726715317057055093335813, 1.06827072941369360132064810231, 2.05375059427311093376821972434, 2.62644611352952404057448947259, 3.70108029534928305640719759984, 4.979145244515386594939851427629, 5.82950640265840986169061838588, 6.33014857656081185772878809041, 6.69343551146725903953022476775, 7.78373243361577709406829102711, 8.31669001228670801922089245790, 8.58175094203350793618164159653, 9.83059741286551874604900910422, 10.35424408959325627417831320835, 10.94567026968108867494823389226, 11.49604629868114514712621641129, 12.818989676875966502625999745391, 13.14133772214725618298095590657, 13.95425204354829611705674774908, 14.37897989935201770786706347799, 15.09744875679549673150797673016, 16.13442308144421479815519264788, 16.873660666091949956866631755664, 17.12598334892500934056901446978, 17.99845672768257499806654813374

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