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Properties

Label 1-23e2-529.491-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (−0.215 + 0.976i)2-s + (0.798 + 0.601i)3-s + (−0.907 − 0.421i)4-s + (0.735 − 0.678i)5-s + (−0.759 + 0.650i)6-s + (−0.287 − 0.957i)7-s + (0.606 − 0.794i)8-s + (0.275 + 0.961i)9-s + (0.503 + 0.863i)10-s + (0.0558 − 0.998i)11-s + (−0.471 − 0.882i)12-s + (−0.982 + 0.185i)13-s + (0.997 − 0.0744i)14-s + (0.995 − 0.0991i)15-s + (0.645 + 0.763i)16-s + (−0.926 − 0.375i)17-s + ⋯
L(s)  = 1  + (−0.215 + 0.976i)2-s + (0.798 + 0.601i)3-s + (−0.907 − 0.421i)4-s + (0.735 − 0.678i)5-s + (−0.759 + 0.650i)6-s + (−0.287 − 0.957i)7-s + (0.606 − 0.794i)8-s + (0.275 + 0.961i)9-s + (0.503 + 0.863i)10-s + (0.0558 − 0.998i)11-s + (−0.471 − 0.882i)12-s + (−0.982 + 0.185i)13-s + (0.997 − 0.0744i)14-s + (0.995 − 0.0991i)15-s + (0.645 + 0.763i)16-s + (−0.926 − 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.431475818 + 0.008501260460i\)
\(L(\frac12)\) \(\approx\) \(1.431475818 + 0.008501260460i\)
\(L(1)\) \(\approx\) \(1.132669851 + 0.2824782840i\)
\(L(1)\) \(\approx\) \(1.132669851 + 0.2824782840i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.215 + 0.976i)T \)
3 \( 1 + (0.798 + 0.601i)T \)
5 \( 1 + (0.735 - 0.678i)T \)
7 \( 1 + (-0.287 - 0.957i)T \)
11 \( 1 + (0.0558 - 0.998i)T \)
13 \( 1 + (-0.982 + 0.185i)T \)
17 \( 1 + (-0.926 - 0.375i)T \)
19 \( 1 + (0.481 - 0.876i)T \)
29 \( 1 + (0.227 + 0.973i)T \)
31 \( 1 + (-0.166 - 0.985i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.813 + 0.581i)T \)
43 \( 1 + (0.299 - 0.954i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (0.503 - 0.863i)T \)
59 \( 1 + (0.879 + 0.476i)T \)
61 \( 1 + (0.0310 + 0.999i)T \)
67 \( 1 + (0.586 - 0.809i)T \)
71 \( 1 + (-0.191 - 0.981i)T \)
73 \( 1 + (0.227 - 0.973i)T \)
79 \( 1 + (-0.0186 + 0.999i)T \)
83 \( 1 + (-0.514 + 0.857i)T \)
89 \( 1 + (-0.999 + 0.0372i)T \)
97 \( 1 + (-0.944 + 0.329i)T \)
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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

−23.20944325139436094771603112581, −22.37631338131115931644862605011, −21.75009824580895037610632866240, −20.86258540591314963629900085198, −20.05036832981993780966997960526, −19.235070662796973544252022378275, −18.61364293192595834257721359931, −17.79913521155291220811411920969, −17.3332401231019854581592361627, −15.53677735329074877021182063296, −14.64605548290694794362866875909, −14.027656426677728561614558498841, −12.911265055669260321455219339000, −12.4611742456625970754162374431, −11.52981555651974063648586337797, −10.129435540521631995144771125637, −9.66131388399103504256096944869, −8.83091615286040961277189030105, −7.77262926505301575571534281021, −6.79365475406942966946482015764, −5.61291211184523065448782603032, −4.25889628558640757609668014471, −2.933935288001278169309356088527, −2.392159965084171230410681171475, −1.60408088878623053603931197583, 0.75155522297631173016811490079, 2.40063060585998210650260370688, 3.84152876539420331813937994032, 4.7039379743676829601009159739, 5.52101834646256684520244359851, 6.81181218025659805015823442136, 7.621977589329934562372715084947, 8.76727640188035621652796383780, 9.26590477744213188223155053383, 10.05301015721231348235984962640, 10.98801274153726345072548875455, 12.8344356796983957538052176788, 13.64219006331443381650047371278, 13.976311192325139299359109813063, 14.98393759605962406990252523880, 16.07421519308904993814938671492, 16.5243580104810655369100815863, 17.283961330612716251700351634198, 18.23221257033824173445656892519, 19.57294902450551425830283403469, 19.85890404992781388454509562006, 21.02519528049894725437372622481, 21.95755109295446838560163650100, 22.46442077266323459998225326327, 24.079501679073147190219447580681

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