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Properties

Label 1-351-351.245-r0-0-0
Degree $1$
Conductor $351$
Sign $-0.482 + 0.875i$
Analytic cond. $1.63003$
Root an. cond. $1.63003$
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  + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.984 − 0.173i)5-s + (−0.342 + 0.939i)7-s + (0.866 − 0.5i)8-s + (0.5 + 0.866i)10-s + (0.342 − 0.939i)11-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)16-s + 17-s + (−0.866 − 0.5i)19-s + (0.342 − 0.939i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (0.939 + 0.342i)25-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.984 − 0.173i)5-s + (−0.342 + 0.939i)7-s + (0.866 − 0.5i)8-s + (0.5 + 0.866i)10-s + (0.342 − 0.939i)11-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)16-s + 17-s + (−0.866 − 0.5i)19-s + (0.342 − 0.939i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (0.939 + 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.482 + 0.875i$
Analytic conductor: \(1.63003\)
Root analytic conductor: \(1.63003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 351,\ (0:\ ),\ -0.482 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07626937737 + 0.1291678124i\)
\(L(\frac12)\) \(\approx\) \(0.07626937737 + 0.1291678124i\)
\(L(1)\) \(\approx\) \(0.4860219101 - 0.09000838108i\)
\(L(1)\) \(\approx\) \(0.4860219101 - 0.09000838108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (-0.342 + 0.939i)T \)
11 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (-0.642 + 0.766i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.342 + 0.939i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.642 + 0.766i)T \)
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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

−24.53404500585955464017857768977, −23.545273011241142037032907763590, −23.13797702078624012877683189565, −22.329137879314316990154850273347, −20.546864270294550836493964630148, −19.99114836800226371593833761771, −19.07882007911882365495030604593, −18.39660136335330503173150795379, −17.080971672144717116134630945862, −16.67774615991809283288361187943, −15.609127852225226038225630538459, −14.829154420670857623360976420735, −14.050464203133011013993668528699, −12.749836902947445483866014129820, −11.65556671755694309252607632324, −10.4565764704423584373974102026, −9.889650466597526175481528988090, −8.588073140477568131101443238846, −7.598776810298950286886594266655, −7.084808613490772366531839838467, −5.95816591251360649739357110800, −4.50952660184709631949312217673, −3.71157929903477318089591123879, −1.749839521277811985902583928840, −0.11718217425971336337228823484, 1.55951634122138867407647871514, 3.06127789779607851380988700873, 3.70709855186205659846707952868, 5.12040814374530550522696675121, 6.55827182118130982559576375892, 7.8360677402611208647804142843, 8.59114033558275840489205681002, 9.33713450705258778842502536098, 10.59730702560417664750268218915, 11.499797863745837445199931301932, 12.194522949360715777292307025004, 12.936535617415036716097950126900, 14.27469853231308585766601982705, 15.53193110852412407523164398611, 16.28064284252570565423769112129, 17.088331574186434620433234049363, 18.41459751651146458563577788895, 18.97152013799726693838834247344, 19.62709304356700802815179398427, 20.531802645822040806655222379838, 21.65107785042009193617399789075, 22.11577422895826737147277939132, 23.29567370076176959910715994041, 24.24394091787526600257899913778, 25.35213424806657337071312796699

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