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Properties

Label 1-29-29.5-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: $\chi_{29} (5, \cdot )$
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

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

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