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Properties

Label 1-53-53.19-r1-0-0
Degree $1$
Conductor $53$
Sign $0.929 + 0.368i$
Analytic cond. $5.69564$
Root an. cond. $5.69564$
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  + (−0.239 − 0.970i)2-s + (0.822 + 0.568i)3-s + (−0.885 + 0.464i)4-s + (−0.935 + 0.354i)5-s + (0.354 − 0.935i)6-s + (0.970 − 0.239i)7-s + (0.663 + 0.748i)8-s + (0.354 + 0.935i)9-s + (0.568 + 0.822i)10-s + (−0.120 + 0.992i)11-s + (−0.992 − 0.120i)12-s + (0.885 + 0.464i)13-s + (−0.464 − 0.885i)14-s + (−0.970 − 0.239i)15-s + (0.568 − 0.822i)16-s + (0.748 + 0.663i)17-s + ⋯
L(s)  = 1  + (−0.239 − 0.970i)2-s + (0.822 + 0.568i)3-s + (−0.885 + 0.464i)4-s + (−0.935 + 0.354i)5-s + (0.354 − 0.935i)6-s + (0.970 − 0.239i)7-s + (0.663 + 0.748i)8-s + (0.354 + 0.935i)9-s + (0.568 + 0.822i)10-s + (−0.120 + 0.992i)11-s + (−0.992 − 0.120i)12-s + (0.885 + 0.464i)13-s + (−0.464 − 0.885i)14-s + (−0.970 − 0.239i)15-s + (0.568 − 0.822i)16-s + (0.748 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.484155230 + 0.2831667266i\)
\(L(\frac12)\) \(\approx\) \(1.484155230 + 0.2831667266i\)
\(L(1)\) \(\approx\) \(1.128539171 + 0.02956628418i\)
\(L(1)\) \(\approx\) \(1.128539171 + 0.02956628418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (-0.239 - 0.970i)T \)
3 \( 1 + (0.822 + 0.568i)T \)
5 \( 1 + (-0.935 + 0.354i)T \)
7 \( 1 + (0.970 - 0.239i)T \)
11 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 + (0.885 + 0.464i)T \)
17 \( 1 + (0.748 + 0.663i)T \)
19 \( 1 + (-0.464 + 0.885i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.120 - 0.992i)T \)
31 \( 1 + (-0.992 + 0.120i)T \)
37 \( 1 + (-0.568 + 0.822i)T \)
41 \( 1 + (0.992 + 0.120i)T \)
43 \( 1 + (-0.568 - 0.822i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
59 \( 1 + (0.354 - 0.935i)T \)
61 \( 1 + (0.663 + 0.748i)T \)
67 \( 1 + (-0.464 - 0.885i)T \)
71 \( 1 + (0.822 - 0.568i)T \)
73 \( 1 + (0.663 - 0.748i)T \)
79 \( 1 + (0.239 - 0.970i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (-0.354 - 0.935i)T \)
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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

−32.90958515499823767821118740738, −31.74911780468904052809389241229, −31.18984702869897840762269264773, −29.895713013140494274949257556, −27.92785556936510864922151002147, −27.220928231361021306165407835863, −26.02800601338508451404849764632, −24.883743983562759077145346490848, −24.00175757223108667129585098065, −23.3105182355116563079310957432, −21.32145722051462857359053686719, −19.90611109136266441562070122471, −18.79440700344167167975478108909, −17.889323139717677602545652726242, −16.22639893680626692939199773015, −15.198322619789590408771185135207, −14.12835931551035429169452287116, −12.932595527163777599604374733391, −11.219161269755043109471919233616, −8.97446332075729520303387437221, −8.22604035279364935149153401719, −7.25939916224252236142849872204, −5.40194958579588520969295715578, −3.64351361769724998092217219305, −0.97195524475240500681181559225, 1.88491193295528326313449867002, 3.662990576012116282036504428980, 4.54032001900224842400278197886, 7.72261213736402110747051116643, 8.5390486766473897450557897552, 10.16349773194809870667455502807, 11.10684964003036200026034849632, 12.49162057721340671180402670289, 14.14145305480229610261215858216, 15.02819048076026079225628188037, 16.68089964912494075086179664931, 18.34799009135139349579940454566, 19.29197287122565244154898336995, 20.53359024222382530470603098999, 21.03803822317812072323837985399, 22.56156672534714315274928378625, 23.679490057182831224885417601675, 25.61124854796349992017399698891, 26.600315935851001808433367640475, 27.53721797177631941926656164740, 28.21964098639654778022552668013, 30.23576648943535951427442184427, 30.77744850874403366499960144918, 31.54717541257565928315518701025, 32.926494586077633202521032046672

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