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Properties

Label 1-1336-1336.123-r0-0-0
Degree $1$
Conductor $1336$
Sign $-0.371 + 0.928i$
Analytic cond. $6.20435$
Root an. cond. $6.20435$
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  + (0.553 + 0.832i)3-s + (−0.584 − 0.811i)5-s + (−0.132 + 0.991i)7-s + (−0.387 + 0.922i)9-s + (0.206 + 0.978i)11-s + (0.997 + 0.0756i)13-s + (0.351 − 0.936i)15-s + (0.993 − 0.113i)17-s + (−0.0944 − 0.995i)19-s + (−0.898 + 0.438i)21-s + (−0.843 + 0.537i)23-s + (−0.316 + 0.948i)25-s + (−0.982 + 0.188i)27-s + (0.843 + 0.537i)29-s + (0.942 + 0.334i)31-s + ⋯
L(s)  = 1  + (0.553 + 0.832i)3-s + (−0.584 − 0.811i)5-s + (−0.132 + 0.991i)7-s + (−0.387 + 0.922i)9-s + (0.206 + 0.978i)11-s + (0.997 + 0.0756i)13-s + (0.351 − 0.936i)15-s + (0.993 − 0.113i)17-s + (−0.0944 − 0.995i)19-s + (−0.898 + 0.438i)21-s + (−0.843 + 0.537i)23-s + (−0.316 + 0.948i)25-s + (−0.982 + 0.188i)27-s + (0.843 + 0.537i)29-s + (0.942 + 0.334i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1336\)    =    \(2^{3} \cdot 167\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(6.20435\)
Root analytic conductor: \(6.20435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1336} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1336,\ (0:\ ),\ -0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8846672834 + 1.306633451i\)
\(L(\frac12)\) \(\approx\) \(0.8846672834 + 1.306633451i\)
\(L(1)\) \(\approx\) \(1.067167549 + 0.4842408625i\)
\(L(1)\) \(\approx\) \(1.067167549 + 0.4842408625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (0.553 + 0.832i)T \)
5 \( 1 + (-0.584 - 0.811i)T \)
7 \( 1 + (-0.132 + 0.991i)T \)
11 \( 1 + (0.206 + 0.978i)T \)
13 \( 1 + (0.997 + 0.0756i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (-0.0944 - 0.995i)T \)
23 \( 1 + (-0.843 + 0.537i)T \)
29 \( 1 + (0.843 + 0.537i)T \)
31 \( 1 + (0.942 + 0.334i)T \)
37 \( 1 + (-0.387 - 0.922i)T \)
41 \( 1 + (-0.776 - 0.629i)T \)
43 \( 1 + (0.800 + 0.599i)T \)
47 \( 1 + (-0.726 + 0.686i)T \)
53 \( 1 + (0.862 - 0.505i)T \)
59 \( 1 + (0.993 + 0.113i)T \)
61 \( 1 + (-0.929 + 0.369i)T \)
67 \( 1 + (0.584 - 0.811i)T \)
71 \( 1 + (-0.169 + 0.985i)T \)
73 \( 1 + (-0.822 + 0.569i)T \)
79 \( 1 + (-0.243 + 0.969i)T \)
83 \( 1 + (-0.988 + 0.150i)T \)
89 \( 1 + (0.351 + 0.936i)T \)
97 \( 1 + (-0.942 + 0.334i)T \)
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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

−20.564256483996408694758220552023, −19.79064163901565345812364539905, −19.002766254798220337461885226096, −18.70514604935965140220304414980, −17.85498305950871757115424105104, −16.85754962464041376369629468246, −16.14804401914094241440173227016, −15.204268898212448806682027702994, −14.2458529736256406181854894663, −13.92641208895398394890180969877, −13.2030761594785762558677577203, −12.04890476827997780383806773827, −11.584596515439432200680696562554, −10.47851351243338046831504383033, −9.98737558756471527930870162448, −8.42631452729796465052937875201, −8.17498471966117847712748405841, −7.31518403501557540740750419652, −6.39531346723742052647678090555, −5.98582472253468096984486884510, −4.18864540151415260343181197593, −3.50754258484987435660266937584, −2.93674402548364120327723258171, −1.56018284031798812735123887849, −0.61770680295541071379804402915, 1.33337101278111179074685677528, 2.45884749873930959717923714580, 3.43293066477356039872370969382, 4.244590403581502776463935728926, 5.03801777042993660485711988522, 5.75342433465474472267555097396, 7.04719186233424144358186039427, 8.13053529897577902438459649643, 8.63239107627674702636457303335, 9.372147819090213747326709858, 10.02548870932684867147134588011, 11.13725234725249360035517248930, 11.93723510442939505097719977526, 12.57380468094802172620578105152, 13.52798984103825290484585695001, 14.40859008661699332345893570158, 15.245141401893676182481909152111, 15.86820136010750117926655621761, 16.16682409480185864624405272438, 17.30920478724634684103537756983, 18.083347140546263002435115282386, 19.20783822240910701513751968625, 19.65004628197191018778361389945, 20.45952728803997354581486279599, 21.13575855753264090206271233451

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