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Properties

Label 1-4011-4011.494-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.616 - 0.787i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.360 + 0.932i)2-s + (−0.739 + 0.673i)4-s + (0.592 + 0.805i)5-s + (−0.894 − 0.446i)8-s + (−0.537 + 0.843i)10-s + (−0.821 + 0.569i)11-s + (−0.956 − 0.293i)13-s + (0.0935 − 0.995i)16-s + (0.609 + 0.792i)17-s + (0.537 + 0.843i)19-s + (−0.980 − 0.197i)20-s + (−0.828 − 0.560i)22-s + (−0.930 − 0.366i)23-s + (−0.298 + 0.954i)25-s + (−0.0715 − 0.997i)26-s + ⋯
L(s)  = 1  + (0.360 + 0.932i)2-s + (−0.739 + 0.673i)4-s + (0.592 + 0.805i)5-s + (−0.894 − 0.446i)8-s + (−0.537 + 0.843i)10-s + (−0.821 + 0.569i)11-s + (−0.956 − 0.293i)13-s + (0.0935 − 0.995i)16-s + (0.609 + 0.792i)17-s + (0.537 + 0.843i)19-s + (−0.980 − 0.197i)20-s + (−0.828 − 0.560i)22-s + (−0.930 − 0.366i)23-s + (−0.298 + 0.954i)25-s + (−0.0715 − 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.616 - 0.787i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (494, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.616 - 0.787i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01507025148 + 0.007335258353i\)
\(L(\frac12)\) \(\approx\) \(0.01507025148 + 0.007335258353i\)
\(L(1)\) \(\approx\) \(0.7081191381 + 0.6342350121i\)
\(L(1)\) \(\approx\) \(0.7081191381 + 0.6342350121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.360 + 0.932i)T \)
5 \( 1 + (0.592 + 0.805i)T \)
11 \( 1 + (-0.821 + 0.569i)T \)
13 \( 1 + (-0.956 - 0.293i)T \)
17 \( 1 + (0.609 + 0.792i)T \)
19 \( 1 + (0.537 + 0.843i)T \)
23 \( 1 + (-0.930 - 0.366i)T \)
29 \( 1 + (0.115 - 0.993i)T \)
31 \( 1 + (-0.137 + 0.990i)T \)
37 \( 1 + (-0.137 - 0.990i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (-0.768 - 0.639i)T \)
47 \( 1 + (-0.982 + 0.186i)T \)
53 \( 1 + (-0.795 - 0.605i)T \)
59 \( 1 + (-0.984 + 0.175i)T \)
61 \( 1 + (0.942 + 0.335i)T \)
67 \( 1 + (-0.997 + 0.0770i)T \)
71 \( 1 + (0.0495 + 0.998i)T \)
73 \( 1 + (0.480 - 0.876i)T \)
79 \( 1 + (-0.917 - 0.396i)T \)
83 \( 1 + (-0.148 + 0.988i)T \)
89 \( 1 + (-0.989 - 0.142i)T \)
97 \( 1 + (-0.999 + 0.0330i)T \)
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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

−18.18162293117336583972198107052, −17.31857704273371330634719027196, −16.568674294898366626498734644911, −15.9117119733453870989310275690, −15.033696766237543263554590063602, −14.12230536758598669115502220555, −13.757757153627923856109337048297, −13.01092870718400527605573056897, −12.51055100298184638033658194832, −11.66285254384061625542902017537, −11.2399636735323784940316444874, −10.06835544179500651161501681476, −9.79456792527941210612385457500, −9.1204865597903537649503975742, −8.28306642001576869428633508408, −7.53297504034119097365095697597, −6.34074424746380050411569133332, −5.585469000969816927430381301588, −4.92180639536405037893647170208, −4.55894925238497279964909899458, −3.26660131469449513959924984828, −2.74084522840550204412904420835, −1.85865280323793918377957623578, −1.03646766592408312565846623633, −0.004184532311780387985348706654, 1.73996739213775097304498963489, 2.58937280337863673997973751681, 3.37443832814420372335182691718, 4.17550367141096910969942704451, 5.18159305918568301179881870407, 5.65545416775769223340155213625, 6.36174687191245095021639101187, 7.20445335527904018431127639020, 7.71469039024798132695212961329, 8.33803239012825800204768331838, 9.46120152127551784793882147097, 10.076059922138195262853879938203, 10.46817902067494927028522000496, 11.736221492663844916414751708690, 12.47602184503400437901942201200, 12.922104717655164017156786417257, 13.92620038226882719391556892412, 14.32077648063288841051841820433, 14.91802908075990751271611713838, 15.56663020597078347449445357299, 16.27864602498366367716848856122, 17.054870216278203592475264012739, 17.71896630655494019362633509136, 18.08115245903869271886394127287, 18.84871966627839789801264679327

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