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Properties

Label 1-152-152.109-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: $\chi_{152} (109, \cdot )$
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

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

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