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Properties

Label 1-34e2-1156.1047-r0-0-0
Degree $1$
Conductor $1156$
Sign $-0.0393 - 0.999i$
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  + (−0.0230 − 0.999i)3-s + (−0.837 + 0.545i)5-s + (−0.424 − 0.905i)7-s + (−0.998 + 0.0461i)9-s + (0.506 + 0.862i)11-s + (−0.183 + 0.982i)13-s + (0.565 + 0.824i)15-s + (0.948 − 0.317i)19-s + (−0.895 + 0.445i)21-s + (−0.905 + 0.424i)23-s + (0.403 − 0.914i)25-s + (0.0692 + 0.997i)27-s + (0.251 − 0.967i)29-s + (−0.206 − 0.978i)31-s + (0.850 − 0.526i)33-s + ⋯
L(s)  = 1  + (−0.0230 − 0.999i)3-s + (−0.837 + 0.545i)5-s + (−0.424 − 0.905i)7-s + (−0.998 + 0.0461i)9-s + (0.506 + 0.862i)11-s + (−0.183 + 0.982i)13-s + (0.565 + 0.824i)15-s + (0.948 − 0.317i)19-s + (−0.895 + 0.445i)21-s + (−0.905 + 0.424i)23-s + (0.403 − 0.914i)25-s + (0.0692 + 0.997i)27-s + (0.251 − 0.967i)29-s + (−0.206 − 0.978i)31-s + (0.850 − 0.526i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0393 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0393 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.0393 - 0.999i$
Analytic conductor: \(5.36844\)
Root analytic conductor: \(5.36844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (0:\ ),\ -0.0393 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6609852594 - 0.6875589227i\)
\(L(\frac12)\) \(\approx\) \(0.6609852594 - 0.6875589227i\)
\(L(1)\) \(\approx\) \(0.7951130312 - 0.2682384818i\)
\(L(1)\) \(\approx\) \(0.7951130312 - 0.2682384818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.0230 - 0.999i)T \)
5 \( 1 + (-0.837 + 0.545i)T \)
7 \( 1 + (-0.424 - 0.905i)T \)
11 \( 1 + (0.506 + 0.862i)T \)
13 \( 1 + (-0.183 + 0.982i)T \)
19 \( 1 + (0.948 - 0.317i)T \)
23 \( 1 + (-0.905 + 0.424i)T \)
29 \( 1 + (0.251 - 0.967i)T \)
31 \( 1 + (-0.206 - 0.978i)T \)
37 \( 1 + (0.160 - 0.986i)T \)
41 \( 1 + (0.723 + 0.690i)T \)
43 \( 1 + (0.486 + 0.873i)T \)
47 \( 1 + (0.673 + 0.739i)T \)
53 \( 1 + (-0.0461 - 0.998i)T \)
59 \( 1 + (0.638 + 0.769i)T \)
61 \( 1 + (0.295 - 0.955i)T \)
67 \( 1 + (0.445 - 0.895i)T \)
71 \( 1 + (0.339 - 0.940i)T \)
73 \( 1 + (-0.620 - 0.783i)T \)
79 \( 1 + (-0.656 - 0.754i)T \)
83 \( 1 + (-0.914 - 0.403i)T \)
89 \( 1 + (-0.183 - 0.982i)T \)
97 \( 1 + (-0.940 - 0.339i)T \)
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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

−21.68560205991651008842490970433, −20.51511966319643389450957193787, −20.12125202806596166130059696553, −19.32811190184964254052355139294, −18.49034517577921801919009930814, −17.48815631675463647223088140859, −16.529227688217928176674561051021, −15.97896624497335611186376790055, −15.54081200872836132174243416368, −14.64397185325413952668202954590, −13.86321310755392138144021720666, −12.57888349084382749589095227337, −12.05486795326536227432552411804, −11.2887696123617658914885171388, −10.38380913447824700578205057960, −9.53597001513323200047576958165, −8.60611585971330704091731318060, −8.34564544627076361623072198437, −7.02148769072690793139037411518, −5.64609997091431045172531803541, −5.41104787970117838923628381426, −4.17899869134839659295996321875, −3.42264379769795613478006811914, −2.71533748410965820026944275165, −0.92476584517792347206408666703, 0.50824640792030149776431984226, 1.74148532002608483246554424540, 2.754559844250677093656265381602, 3.84082370062336096699853799707, 4.49997161545340653870752834708, 6.036199099925596036314431631938, 6.72718011876217944676990587924, 7.51236830461708394746429344226, 7.78701602350566290755568427614, 9.20855348084205429901187915452, 9.9210450818496972683176313289, 11.159452697230370440679092883176, 11.64190303994409305465505201256, 12.393144573729731616423119995827, 13.2732594535401919547802244884, 14.137585271824964066846898431143, 14.57416807418461256276044662431, 15.74122288826258206244430515167, 16.48277559119376804734887373977, 17.37959798042336980089688365422, 18.02077346782114878689994318701, 18.88636709180416352078603864880, 19.70713611936864447131662644695, 19.828405768698535278046221518192, 20.87752422084101935509076068116

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