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Properties

Label 1-153-153.25-r0-0-0
Degree $1$
Conductor $153$
Sign $0.812 - 0.582i$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.965 − 0.258i)7-s + i·8-s + (0.707 − 0.707i)10-s + (0.965 + 0.258i)11-s + (0.5 − 0.866i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s − i·19-s + (−0.258 + 0.965i)20-s + (−0.965 + 0.258i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.5i)25-s + i·26-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.965 − 0.258i)7-s + i·8-s + (0.707 − 0.707i)10-s + (0.965 + 0.258i)11-s + (0.5 − 0.866i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s − i·19-s + (−0.258 + 0.965i)20-s + (−0.965 + 0.258i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.5i)25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.812 - 0.582i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.812 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4971262974 - 0.1598388905i\)
\(L(\frac12)\) \(\approx\) \(0.4971262974 - 0.1598388905i\)
\(L(1)\) \(\approx\) \(0.5857843683 + 0.007878766971i\)
\(L(1)\) \(\approx\) \(0.5857843683 + 0.007878766971i\)

Euler product

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

−27.96885393084219527517859450296, −27.32442208332363168486828372653, −26.30409314096618485229399679227, −25.42263740285166503306552255237, −24.38704656180737481773252810963, −23.10445761972247263689397086931, −22.103454662233752968561288650956, −21.01557182487989099551426029642, −19.84790242945195782990783347478, −19.26695491486551745526544297248, −18.50691091401362822415908029069, −17.00001099016273207441189614849, −16.27077711248044083201055262365, −15.46535277355830419862916281322, −13.78660538082247876111159073512, −12.306226072623917823365916490868, −11.86496089078326857942715657356, −10.62787444233037703531849610494, −9.32320402361006731577217599560, −8.623162931136772806967340696793, −7.32678501844456370370134343186, −6.26221938029541784845017199211, −4.060655320102976490498281528162, −3.232673824944687649965316699503, −1.363019608513072700958748543606, 0.66779190629006784186483363062, 2.84145148152944413590258622807, 4.31452634470539928593913177903, 6.15068703357765815071577814873, 6.96964738163615204781814915504, 8.06945697428103647009865593739, 9.1619113968123028658567573913, 10.29220486728362526170821210083, 11.287309924422762393005102397226, 12.481209264015936883876487739841, 13.99876725247461742403573609047, 15.25384936279688310419737080966, 15.84340811527084921633687073710, 16.88408948373160904475286904029, 17.91606989651803308708512246649, 19.136784676916510699421935118387, 19.65047269870065993455571050195, 20.54287183504637603549960850675, 22.49894229626565295372243603858, 22.991560362407593989277701665192, 24.1702777295243294782449342158, 25.13030707255309134440933403395, 26.13317081990835385236258908289, 26.81930422389615333320894578660, 27.87487938677039258855273351321

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