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Properties

Label 1-34e2-1156.171-r1-0-0
Degree $1$
Conductor $1156$
Sign $0.345 - 0.938i$
Analytic cond. $124.229$
Root an. cond. $124.229$
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  + (−0.932 − 0.361i)3-s + (−0.982 + 0.183i)5-s + (−0.739 − 0.673i)7-s + (0.739 + 0.673i)9-s + (0.602 − 0.798i)11-s + (−0.982 − 0.183i)13-s + (0.982 + 0.183i)15-s + (−0.445 + 0.895i)19-s + (0.445 + 0.895i)21-s + (−0.739 − 0.673i)23-s + (0.932 − 0.361i)25-s + (−0.445 − 0.895i)27-s + (−0.602 + 0.798i)29-s + (0.982 + 0.183i)31-s + (−0.850 + 0.526i)33-s + ⋯
L(s)  = 1  + (−0.932 − 0.361i)3-s + (−0.982 + 0.183i)5-s + (−0.739 − 0.673i)7-s + (0.739 + 0.673i)9-s + (0.602 − 0.798i)11-s + (−0.982 − 0.183i)13-s + (0.982 + 0.183i)15-s + (−0.445 + 0.895i)19-s + (0.445 + 0.895i)21-s + (−0.739 − 0.673i)23-s + (0.932 − 0.361i)25-s + (−0.445 − 0.895i)27-s + (−0.602 + 0.798i)29-s + (0.982 + 0.183i)31-s + (−0.850 + 0.526i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(124.229\)
Root analytic conductor: \(124.229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (1:\ ),\ 0.345 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3927234082 - 0.2737526158i\)
\(L(\frac12)\) \(\approx\) \(0.3927234082 - 0.2737526158i\)
\(L(1)\) \(\approx\) \(0.5259732220 - 0.08770165190i\)
\(L(1)\) \(\approx\) \(0.5259732220 - 0.08770165190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.932 - 0.361i)T \)
5 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (-0.739 - 0.673i)T \)
11 \( 1 + (0.602 - 0.798i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
19 \( 1 + (-0.445 + 0.895i)T \)
23 \( 1 + (-0.739 - 0.673i)T \)
29 \( 1 + (-0.602 + 0.798i)T \)
31 \( 1 + (0.982 + 0.183i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (0.932 + 0.361i)T \)
43 \( 1 + (0.273 - 0.961i)T \)
47 \( 1 + (-0.739 + 0.673i)T \)
53 \( 1 + (0.739 + 0.673i)T \)
59 \( 1 + (-0.0922 + 0.995i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (-0.445 + 0.895i)T \)
71 \( 1 + (-0.739 - 0.673i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (-0.445 + 0.895i)T \)
83 \( 1 + (-0.932 + 0.361i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (0.739 + 0.673i)T \)
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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

−21.45639361283271426169967327207, −20.43647963694176571885988173252, −19.48094689082570209485543189791, −19.16669450449400498938689531471, −18.02085614976657297696144298814, −17.26800555423190959641565040822, −16.636564385032353786010897129221, −15.61105996930732691353870411262, −15.4352175966636626693221963382, −14.51799425622443076764831151888, −13.06200884353674391684669735730, −12.41645805336013726490395556903, −11.77976446200417011891910281980, −11.297997052935605358123134970444, −9.966015230761100636340088153677, −9.58222074462550215165717628167, −8.56404848662901668156064797872, −7.359517702642436676427989062376, −6.75054589963315253056515738245, −5.83414649628690358494953546468, −4.78247652669235356778616725336, −4.23772587913158707270265354074, −3.20851828010509079450643567726, −1.94136878913869726767139250405, −0.43901174652467984686900043464, 0.2714397196015737971552744930, 1.21607106097454131607055441239, 2.7150098209953825856735393874, 3.83957472720222459178828476394, 4.42560381007868605004504072298, 5.66065771863553228886611204025, 6.463307658846096694226424854971, 7.200289969600600146034616548138, 7.86100620393381457060031406943, 8.937839292255474316181940390649, 10.25017140748484439644645903166, 10.59092979382432189800677621712, 11.63710458934466354483145282545, 12.2379631237365411610208725996, 12.84097315269676009275364403463, 13.92022749244080595670231495006, 14.683526154428021199037509478727, 15.791371929485205977780998560832, 16.47178804514211780413627114279, 16.84976882219761887221841539125, 17.82353376388594953802192559967, 18.79026335762258234225791058216, 19.36173135833307920281562384857, 19.8386899188463526930209974218, 20.97467773547300838014355274122

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