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Properties

Label 1-53-53.30-r1-0-0
Degree $1$
Conductor $53$
Sign $-0.138 - 0.990i$
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  + i·2-s + i·3-s − 4-s − i·5-s − 6-s − 7-s − i·8-s − 9-s + 10-s − 11-s − i·12-s + 13-s − i·14-s + 15-s + 16-s − 17-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − i·5-s − 6-s − 7-s − i·8-s − 9-s + 10-s − 11-s − i·12-s + 13-s − i·14-s + 15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ -0.138 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05822670413 + 0.06694993833i\)
\(L(\frac12)\) \(\approx\) \(-0.05822670413 + 0.06694993833i\)
\(L(1)\) \(\approx\) \(0.4872137428 + 0.3675060145i\)
\(L(1)\) \(\approx\) \(0.4872137428 + 0.3675060145i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 - iT \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 + T \)
47 \( 1 - iT \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - iT \)
71 \( 1 + iT \)
73 \( 1 + iT \)
79 \( 1 - iT \)
83 \( 1 - iT \)
89 \( 1 - iT \)
97 \( 1 + T \)
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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

−31.708217706019056058430204414953, −30.89333467492877942902613697700, −29.90758722180024823803591293233, −29.10831196185425817313501019915, −28.16636347211122656835479846753, −26.29475958793087016464684236243, −25.83014042204295008026253268722, −23.79813874454838312892339623442, −22.89562409876483683822729199029, −21.97627854575145607140205110536, −20.35382240087982279916323108389, −19.19668316759914238621555708091, −18.510262675939968690915797347214, −17.52921108137947601946757672100, −15.41713964201694600536646809048, −13.57785116042492632378093680758, −13.17134465316881975164155395236, −11.56302910755392008928514290877, −10.57791449423545993649850753439, −8.97716767125713724145370799659, −7.322496326398798378142693672595, −5.86522103953432097424505888091, −3.41051091464846208028125855990, −2.25783551405326456195839795228, −0.04805049208372947696732306008, 3.68829551351346945624833233241, 5.01440859342014329621987694858, 6.20192201193967662288434994607, 8.27397546470600865440812561016, 9.1949323527045135377458038216, 10.43646034649989825509627515526, 12.65358539153539198848514923544, 13.72598832738135104596123076320, 15.43420969867392313235615125119, 16.10449215835529591756023539884, 16.89618330293627451926556909447, 18.45369557804992667431575863413, 20.13099862977190108162549794313, 21.285545300483289901284192931459, 22.609710465107428010475282102698, 23.49362324876369883754916441660, 24.93279554017313311377643307692, 25.95747202937140892521635325719, 26.81522852557843769667957672230, 28.173915173817873533556323520813, 28.78756159447820815858309135173, 31.22784986395769305698882641392, 31.92737002806216780008758942507, 32.860610550760393952346212830998, 33.566741492190754812000200677364

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