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Properties

Label 1-155-155.29-r1-0-0
Degree $1$
Conductor $155$
Sign $-0.943 + 0.330i$
Analytic cond. $16.6570$
Root an. cond. $16.6570$
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  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(16.6570\)
Root analytic conductor: \(16.6570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (1:\ ),\ -0.943 + 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2425492728 - 1.425485991i\)
\(L(\frac12)\) \(\approx\) \(-0.2425492728 - 1.425485991i\)
\(L(1)\) \(\approx\) \(0.5744166844 - 0.8299658705i\)
\(L(1)\) \(\approx\) \(0.5744166844 - 0.8299658705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−27.97877244003275214079503624687, −27.210859829530566302233341669268, −26.2151722015609439997111330959, −25.530778288141503541381524976823, −24.46389061006345600710769989671, −23.60371606219254448961171964909, −22.15997606150971375290716958998, −21.74238303460021682881784469421, −20.28502839577315882693828969275, −19.34396127940623328189617667259, −18.005434730998032935452046823744, −17.20515602338844306899933923592, −16.119393139741101341467461922400, −15.22161237510550324615948874792, −14.53138195215559279526110616007, −13.58311425344685968114598640553, −11.74295535824939694107971635767, −10.60863284452914757546923720977, −9.17532853909970942778939081513, −8.86505050181775043654694352105, −7.46697314296573839330406732966, −6.08663150178332181734981481547, −4.84926062815347156708098241789, −4.01912415743328866605130310850, −1.86287316836622699951883417953, 0.60812240350539105913345652216, 1.661836768770507839482717526301, 3.04393023363099573386591656180, 4.32650034404355614006056939951, 6.08191311167047506873675249651, 7.71624012850341001718426869255, 8.3090020904233716702001799624, 9.586638170050329208437972948, 11.012882600350630100365590099195, 11.72280931846166062347297830133, 12.88613442461796976510734681803, 13.80238137287565842909550123638, 14.56572054092430997318863372090, 16.5592816581076705325711633633, 17.68732437225938567974344811843, 18.214490126348016971546129838980, 19.36198555076603814255499507561, 20.213498284557253777328465658548, 20.83385816532721131637654366482, 22.296122231587487153743147498845, 23.11590079905224106430567722772, 24.34212237183541930177243318137, 25.14609301142868815841355884752, 26.490394699890388532680966720272, 27.17485905729137935085595163502

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