Properties

Label 1-152-152.53-r1-0-0
Degree $1$
Conductor $152$
Sign $0.837 + 0.546i$
Analytic cond. $16.3346$
Root an. cond. $16.3346$
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  + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)17-s + (0.766 + 0.642i)21-s + (0.173 + 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.766 − 0.642i)29-s + (0.5 + 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.766 + 0.642i)17-s + (0.766 + 0.642i)21-s + (0.173 + 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)27-s + (0.766 − 0.642i)29-s + (0.5 + 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044191251 + 0.3103067542i\)
\(L(\frac12)\) \(\approx\) \(1.044191251 + 0.3103067542i\)
\(L(1)\) \(\approx\) \(0.7845985498 + 0.1204456933i\)
\(L(1)\) \(\approx\) \(0.7845985498 + 0.1204456933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−27.93869012157282636862511943201, −27.07802521729647197733939514631, −25.34908253710874858539333895020, −24.76335042469911181475161474191, −23.81425222759948887363353814267, −22.76836629604105999024921273733, −22.01279732199714574264484630994, −20.86532258069135228217828008891, −19.645290816239650026723332503326, −18.728265827917823572960126368435, −17.59743179814922440796353142116, −16.67948573414820532721752156265, −15.955151014003575468714421620419, −14.662961435220775382204367349369, −13.04409558056144311877821105394, −12.226226993989448481583600394412, −11.78234347319224253615721120559, −10.03574316001458031735946816225, −9.144840866684205846246960011591, −7.673577279819604948702747619968, −6.49315350168174676671008028455, −5.29082402330210032352219911344, −4.42713543632337765144937151930, −2.28245963795599916774408950824, −0.71894953741166340181681436746, 0.826905032179019348688044281317, 3.136608353783755189058563372219, 4.16877303198374444656195845441, 5.75697239817774491925347260139, 6.69013904098994604544871112228, 7.71424457245501720906144140743, 9.67676011712995750019752937641, 10.43235509537292874987736248640, 11.3272306750531809077850036042, 12.38264992231876305021336195055, 13.76455073391979431079469242745, 14.83914910325250859002135991944, 15.95006087760327570844818910918, 16.95011637754554835531095708349, 17.685915407687269416431354646401, 19.01620335025025117144351258417, 19.723340644313877686345037027035, 21.39160260814702673611739475163, 22.016476110462824192927001278132, 23.048004999673056768451114956894, 23.55210458767010084565351552769, 24.91414344797137795552912887308, 26.305942747035982788457226762371, 26.91820421859573987350631756597, 27.68330307013542858756692388114

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