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Properties

Label 1-2527-2527.1095-r0-0-0
Degree $1$
Conductor $2527$
Sign $0.976 + 0.216i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (−0.0275 − 0.999i)2-s + (−0.986 + 0.164i)3-s + (−0.998 + 0.0550i)4-s + (−0.993 + 0.110i)5-s + (0.191 + 0.981i)6-s + (0.0825 + 0.996i)8-s + (0.945 − 0.324i)9-s + (0.137 + 0.990i)10-s + (−0.191 − 0.981i)11-s + (0.975 − 0.218i)12-s + (0.635 + 0.771i)13-s + (0.962 − 0.272i)15-s + (0.993 − 0.110i)16-s + (−0.546 + 0.837i)17-s + (−0.350 − 0.936i)18-s + ⋯
L(s)  = 1  + (−0.0275 − 0.999i)2-s + (−0.986 + 0.164i)3-s + (−0.998 + 0.0550i)4-s + (−0.993 + 0.110i)5-s + (0.191 + 0.981i)6-s + (0.0825 + 0.996i)8-s + (0.945 − 0.324i)9-s + (0.137 + 0.990i)10-s + (−0.191 − 0.981i)11-s + (0.975 − 0.218i)12-s + (0.635 + 0.771i)13-s + (0.962 − 0.272i)15-s + (0.993 − 0.110i)16-s + (−0.546 + 0.837i)17-s + (−0.350 − 0.936i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1095, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ 0.976 + 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5180347665 + 0.05672360394i\)
\(L(\frac12)\) \(\approx\) \(0.5180347665 + 0.05672360394i\)
\(L(1)\) \(\approx\) \(0.5285596181 - 0.1900036826i\)
\(L(1)\) \(\approx\) \(0.5285596181 - 0.1900036826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.0275 - 0.999i)T \)
3 \( 1 + (-0.986 + 0.164i)T \)
5 \( 1 + (-0.993 + 0.110i)T \)
11 \( 1 + (-0.191 - 0.981i)T \)
13 \( 1 + (0.635 + 0.771i)T \)
17 \( 1 + (-0.546 + 0.837i)T \)
23 \( 1 + (-0.986 - 0.164i)T \)
29 \( 1 + (0.998 + 0.0550i)T \)
31 \( 1 + (0.851 + 0.523i)T \)
37 \( 1 + (0.754 - 0.656i)T \)
41 \( 1 + (0.716 + 0.697i)T \)
43 \( 1 + (-0.926 + 0.376i)T \)
47 \( 1 + (-0.945 - 0.324i)T \)
53 \( 1 + (0.754 - 0.656i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (0.0825 - 0.996i)T \)
67 \( 1 + (-0.904 - 0.426i)T \)
71 \( 1 + (-0.904 - 0.426i)T \)
73 \( 1 + (-0.546 + 0.837i)T \)
79 \( 1 + (-0.137 + 0.990i)T \)
83 \( 1 + (0.401 + 0.915i)T \)
89 \( 1 + (0.546 + 0.837i)T \)
97 \( 1 + (-0.821 - 0.569i)T \)
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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

−19.176213051693410612574628161527, −18.36121200855595149132268199481, −17.86781414061009653136660585989, −17.37165620960110888015648218599, −16.276340736631859433919223688924, −16.019362679381020705419580877691, −15.34214467283988667041808759067, −14.76627427303288424059340269962, −13.46449594816060134417325799330, −13.144946645465407980345319630225, −12.018676054976479744005379252183, −11.83783763854392790782916984620, −10.566459766460962490957447911774, −10.08462810612067016893794444931, −9.05587643948397786243643201584, −8.10264047264521590377965306112, −7.58388850837219139104567979453, −6.8913187557331302305139918739, −6.158038812277712298243278944610, −5.344749110878092949644668368274, −4.48259448955418183334609389199, −4.183636464133213566632062466576, −2.859866678116737807224441545492, −1.28737717271430532799486716737, −0.32924290983078206663890925109, 0.7335238893032715252996087734, 1.64127849398227551730169597723, 2.88627110388780563212199959615, 3.8260884290155218404812971306, 4.25989377963875129294054627530, 5.07434512040569034928973268074, 6.12038780597233453109237627740, 6.705996199751918090993534890754, 8.10116050646518706163910480677, 8.41042010649315708706543015794, 9.48078344972906557356427656227, 10.35839668219380430270814727604, 10.97157941691001232027979079837, 11.457634752083680179010164624738, 12.013341286357103426147551179780, 12.76364880857886180927089539494, 13.49348808482084631978017257541, 14.352053719233350494079666515436, 15.25165075567694594014214550629, 16.26355289266727989947516947323, 16.39445206367106948083763853107, 17.585854423696291312286624319607, 18.129962634148193060019417818541, 18.82162550272428543580243990522, 19.43544300473392269247235556568

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