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Properties

Label 1-29-29.6-r0-0-0
Degree $1$
Conductor $29$
Sign $0.357 + 0.934i$
Analytic cond. $0.134675$
Root an. cond. $0.134675$
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  + (0.222 + 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.623 + 0.781i)11-s − 12-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)14-s + (0.222 − 0.974i)15-s + (0.623 − 0.781i)16-s − 17-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.900 − 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.623 + 0.781i)11-s − 12-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)14-s + (0.222 − 0.974i)15-s + (0.623 − 0.781i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.357 + 0.934i$
Analytic conductor: \(0.134675\)
Root analytic conductor: \(0.134675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (0:\ ),\ 0.357 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7180344568 + 0.4941222572i\)
\(L(\frac12)\) \(\approx\) \(0.7180344568 + 0.4941222572i\)
\(L(1)\) \(\approx\) \(0.9788579405 + 0.5013789824i\)
\(L(1)\) \(\approx\) \(0.9788579405 + 0.5013789824i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 - T \)
19 \( 1 + (0.900 - 0.433i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.222 + 0.974i)T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 - T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (-0.623 - 0.781i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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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

−37.472816684805367884766126485304, −36.05290985367418708932697838197, −34.9779251993863654677059534847, −32.94398106367929238881590275443, −31.53535246510824253850563607235, −30.95906036914419741131251761688, −29.67268131011723952403343421659, −28.713235718478163069382242737680, −26.73503824687056749258469337023, −26.086393307424357170969489348231, −24.19503769002546292200700747881, −22.75413382232820657565328128440, −21.56927833632389005558957901429, −20.14682615061081844471041391670, −18.8345936296567753203993983291, −18.4745283405985560358470105866, −15.6099313237819781463759453578, −14.12623375479326874022967911496, −13.14389844034692319456445844442, −11.55120507413995855662053352753, −9.94970203852027398577326840882, −8.50501843972967509078698302643, −6.416861012020751427602064281908, −3.62705518518547117294005850178, −2.49079792248403447627084129787, 3.60892371717206461070268744899, 5.08758665309842941298914414390, 7.31429291405065213960582874952, 8.65478881611810207486834029951, 9.8614053478242136375450757624, 12.84823028408343441070384092705, 13.62237875743375831166492554543, 15.54978997072314262633148132905, 16.00049400663150968222210935604, 17.70425038990496369366756061426, 19.64337154397940999991294791273, 20.752431183315054086889118907363, 22.38363900482181567389609371500, 23.72872326294396064186236822911, 25.07386857160635349587185482185, 25.9351440459263721803012624346, 27.107608237767454243147265172099, 28.454218548616423298675408796222, 30.64458272977439180965131044197, 31.68982615438367985291580467558, 32.614478472899796839997785977164, 33.36464701017846370341614943169, 35.39090313652722092796549573143, 36.00013704825838537402679077079, 37.21774510218629940317127496853

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