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Properties

Label 1-34e2-1156.335-r0-0-0
Degree $1$
Conductor $1156$
Sign $-0.0570 - 0.998i$
Analytic cond. $5.36844$
Root an. cond. $5.36844$
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  + (0.993 + 0.115i)3-s + (−0.251 − 0.967i)5-s + (0.584 − 0.811i)7-s + (0.973 + 0.228i)9-s + (−0.884 + 0.466i)11-s + (−0.798 − 0.602i)13-s + (−0.138 − 0.990i)15-s + (0.0461 − 0.998i)19-s + (0.673 − 0.739i)21-s + (0.811 + 0.584i)23-s + (−0.873 + 0.486i)25-s + (0.940 + 0.339i)27-s + (0.295 − 0.955i)29-s + (0.506 + 0.862i)31-s + (−0.932 + 0.361i)33-s + ⋯
L(s)  = 1  + (0.993 + 0.115i)3-s + (−0.251 − 0.967i)5-s + (0.584 − 0.811i)7-s + (0.973 + 0.228i)9-s + (−0.884 + 0.466i)11-s + (−0.798 − 0.602i)13-s + (−0.138 − 0.990i)15-s + (0.0461 − 0.998i)19-s + (0.673 − 0.739i)21-s + (0.811 + 0.584i)23-s + (−0.873 + 0.486i)25-s + (0.940 + 0.339i)27-s + (0.295 − 0.955i)29-s + (0.506 + 0.862i)31-s + (−0.932 + 0.361i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0570 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0570 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.0570 - 0.998i$
Analytic conductor: \(5.36844\)
Root analytic conductor: \(5.36844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (0:\ ),\ -0.0570 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329181100 - 1.407288071i\)
\(L(\frac12)\) \(\approx\) \(1.329181100 - 1.407288071i\)
\(L(1)\) \(\approx\) \(1.319524532 - 0.4541435691i\)
\(L(1)\) \(\approx\) \(1.319524532 - 0.4541435691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.993 + 0.115i)T \)
5 \( 1 + (-0.251 - 0.967i)T \)
7 \( 1 + (0.584 - 0.811i)T \)
11 \( 1 + (-0.884 + 0.466i)T \)
13 \( 1 + (-0.798 - 0.602i)T \)
19 \( 1 + (0.0461 - 0.998i)T \)
23 \( 1 + (0.811 + 0.584i)T \)
29 \( 1 + (0.295 - 0.955i)T \)
31 \( 1 + (0.506 + 0.862i)T \)
37 \( 1 + (0.690 - 0.723i)T \)
41 \( 1 + (-0.620 - 0.783i)T \)
43 \( 1 + (-0.565 - 0.824i)T \)
47 \( 1 + (-0.526 + 0.850i)T \)
53 \( 1 + (0.228 - 0.973i)T \)
59 \( 1 + (0.317 - 0.948i)T \)
61 \( 1 + (-0.0692 + 0.997i)T \)
67 \( 1 + (0.739 - 0.673i)T \)
71 \( 1 + (-0.160 - 0.986i)T \)
73 \( 1 + (-0.978 - 0.206i)T \)
79 \( 1 + (0.905 + 0.424i)T \)
83 \( 1 + (-0.486 - 0.873i)T \)
89 \( 1 + (-0.798 + 0.602i)T \)
97 \( 1 + (0.986 - 0.160i)T \)
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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

−21.548731498560552342276001775898, −20.74992259149725800401378502630, −19.89324353843314505802303327827, −18.89088978124459881742346470711, −18.659941393652542485646853446443, −18.03770642977203065209286629432, −16.75165907498825170354996837163, −15.836642429185190014133908220256, −14.89188441548532142209453955551, −14.74767073369792360960798457737, −13.86470530876224436967742401050, −12.97438286583965396957260935044, −12.05181685996520942982900101685, −11.281505364655985788173866528621, −10.292520423850144510218298190521, −9.620286095970947304729933393468, −8.48707774065617795666273418173, −8.02717947649731338438476836955, −7.15964215430844598700908689935, −6.30219563411001376938967494884, −5.12973760662544931942413273990, −4.18335436758546384940505893312, −2.95638873832926670412402477562, −2.62026194078496456419033594119, −1.55295804406162942397744027932, 0.67670219211141608478184974344, 1.83335935249647811016818848885, 2.80015565699981124691810390760, 3.86202012923236233288855218281, 4.824847276372304859290978352788, 5.13729237359374776140470501100, 6.9523544718694315617757469395, 7.65900902165198528919942081311, 8.18048302051238064918762043546, 9.08950037354030539877625390975, 9.905104555309458070782761732052, 10.623057140814805808535233521720, 11.72018441439614469858839164531, 12.75259075500791511477143844565, 13.259055054108025331298267944884, 13.96345438425303693845945278628, 15.020020859164308262695455599781, 15.488008946832570969263834906239, 16.320690422278769106204084421682, 17.37466739674296191328621002851, 17.78054909446546166764045123994, 19.078207289551427158264349818015, 19.76989064036929511282732936333, 20.23993624585055012440446959471, 21.0108564131776672094838783797

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