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Properties

Label 1-161-161.117-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.931 + 0.363i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (0.580 − 0.814i)2-s + (−0.235 + 0.971i)3-s + (−0.327 − 0.945i)4-s + (−0.0475 + 0.998i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (−0.888 − 0.458i)9-s + (0.786 + 0.618i)10-s + (0.580 + 0.814i)11-s + (0.995 − 0.0950i)12-s + (0.142 − 0.989i)13-s + (−0.959 − 0.281i)15-s + (−0.786 + 0.618i)16-s + (−0.981 + 0.189i)17-s + (−0.888 + 0.458i)18-s + (−0.981 − 0.189i)19-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.235 + 0.971i)3-s + (−0.327 − 0.945i)4-s + (−0.0475 + 0.998i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (−0.888 − 0.458i)9-s + (0.786 + 0.618i)10-s + (0.580 + 0.814i)11-s + (0.995 − 0.0950i)12-s + (0.142 − 0.989i)13-s + (−0.959 − 0.281i)15-s + (−0.786 + 0.618i)16-s + (−0.981 + 0.189i)17-s + (−0.888 + 0.458i)18-s + (−0.981 − 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.931 + 0.363i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.931 + 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05799710300 + 0.3083223169i\)
\(L(\frac12)\) \(\approx\) \(0.05799710300 + 0.3083223169i\)
\(L(1)\) \(\approx\) \(0.9087416500 + 4.633020717\times10^{-5}i\)
\(L(1)\) \(\approx\) \(0.9087416500 + 4.633020717\times10^{-5}i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.580 - 0.814i)T \)
3 \( 1 + (-0.235 + 0.971i)T \)
5 \( 1 + (-0.0475 + 0.998i)T \)
11 \( 1 + (0.580 + 0.814i)T \)
13 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (-0.981 + 0.189i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (-0.723 + 0.690i)T \)
37 \( 1 + (-0.888 - 0.458i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.928 + 0.371i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.327 + 0.945i)T \)
79 \( 1 + (0.928 - 0.371i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−26.999783740353581336496197807341, −25.748830946996553925839899559363, −24.91862572511620814998786676634, −24.010379430076695771987477603771, −23.773963477697609166286718854414, −22.43631354123575222089365424346, −21.51069708269162693981696601851, −20.28586617606925572709963508208, −19.12871678044855851645861162438, −18.01499936543409078016222370963, −16.72121530121950322940805303182, −16.63270514890484720254028345026, −15.03070735343616493021840495567, −13.7642524273296828835516634066, −13.218715140793596626506932388583, −12.1434150871390619631723192507, −11.34704889840550724027998393309, −8.9459414478989949780639308998, −8.450539926535296915864512126048, −7.05042118985681355785900319486, −6.18650551667205287639370757270, −5.07028581738440689303316512628, −3.80705594040924683300640337302, −1.88438608153628272586554823873, −0.09186006461360882276648228828, 2.203232617761227414567510400852, 3.46275416517573448314583066015, 4.361978962828120547859216239946, 5.6639649621908525418392957233, 6.78357577352587783488893963135, 8.80373773750511412641762959318, 9.99717980237771323664735793204, 10.691921897082617076610610203492, 11.50493352314657754914554661434, 12.71431371079358674979297944620, 14.05117006650456391727584207862, 15.090581048325853825798916007296, 15.42196868209434490231612634651, 17.282549498883707484863632345922, 18.12213438968588198545672467530, 19.486276813618444208018836647749, 20.2365188151549250597019438379, 21.292444482095222071950626452561, 22.24669277537523780677269891767, 22.67110754703729802424369073294, 23.57885573590993911448225505990, 25.16002580696750096131478867172, 26.27266986965110683750786503184, 27.31358031041326792258444785799, 27.918126814905210050214910231524

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