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Properties

Label 1-153-153.113-r0-0-0
Degree $1$
Conductor $153$
Sign $0.998 + 0.0524i$
Analytic cond. $0.710529$
Root an. cond. $0.710529$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (−0.608 + 0.793i)5-s + (0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (−0.382 + 0.923i)10-s + (0.793 − 0.608i)11-s + (−0.866 + 0.5i)13-s + (0.793 + 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (−0.130 + 0.991i)20-s + (0.608 − 0.793i)22-s + (−0.130 − 0.991i)23-s + (−0.258 − 0.965i)25-s + (−0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (−0.608 + 0.793i)5-s + (0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (−0.382 + 0.923i)10-s + (0.793 − 0.608i)11-s + (−0.866 + 0.5i)13-s + (0.793 + 0.608i)14-s + (0.5 − 0.866i)16-s + (0.707 + 0.707i)19-s + (−0.130 + 0.991i)20-s + (0.608 − 0.793i)22-s + (−0.130 − 0.991i)23-s + (−0.258 − 0.965i)25-s + (−0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.998 + 0.0524i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.998 + 0.0524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.892746454 + 0.04963967992i\)
\(L(\frac12)\) \(\approx\) \(1.892746454 + 0.04963967992i\)
\(L(1)\) \(\approx\) \(1.719164702 + 0.02322649849i\)
\(L(1)\) \(\approx\) \(1.719164702 + 0.02322649849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (0.608 + 0.793i)T \)
11 \( 1 + (0.793 - 0.608i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.130 - 0.991i)T \)
29 \( 1 + (-0.991 - 0.130i)T \)
31 \( 1 + (-0.793 - 0.608i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (0.991 - 0.130i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (-0.608 - 0.793i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.793 + 0.608i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.991 + 0.130i)T \)
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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

−27.92295690842825786798446551591, −27.08278556194079976733262420749, −25.853395627990536105677997582346, −24.627971951361781977189639116253, −24.15236264223298291831063559158, −23.16829433142092456464903106573, −22.29760579756160613345770640428, −21.116694418443789081404043330895, −20.05629557875576818881243265835, −19.76395563405594536346681271985, −17.567242486335540310718185163523, −16.92944010905464834698271421067, −15.79613366590109988257960876053, −14.84985484314678992556833695182, −13.875202622401150210550399843162, −12.73916835674180835003637278763, −11.89640491052029678093738542457, −10.92258757517762678020273777823, −9.27621647540461816853304376550, −7.70722771788036653379175891338, −7.18018382841953006223443870928, −5.37482104307194032081928219160, −4.53268475920231481782954536315, −3.50681417015983469530627311512, −1.595156082277204614763387968412, 1.949153501974844449748544472168, 3.200516664282943840399173275557, 4.353151386583377251958355469106, 5.67634135369152811870515151170, 6.78700983193952210881554193469, 7.96592440276973821277775056287, 9.59952171685145576302786565549, 11.05331729986405319613981358617, 11.67606203060639816120425671305, 12.55550338085196633903377128703, 14.3129087540426886791079455916, 14.514394319244717249957518720749, 15.66732395759608752992478110235, 16.763120083582548463701127032946, 18.455371441115928925314371843692, 19.156191688794401557504829428091, 20.20945141248855433204637934623, 21.39772920481707034804974228231, 22.2141137198530774592720913844, 22.8268904456344359477842440609, 24.337440075801460371486163602476, 24.49970733569007617179147779082, 26.012392294914535743183101021489, 27.17206167562399693826988626528, 28.0471060552533545150201461930

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