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Properties

Label 2-172-172.7-c1-0-11
Degree $2$
Conductor $172$
Sign $-0.0162 + 0.999i$
Analytic cond. $1.37342$
Root an. cond. $1.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.903 − 1.08i)2-s + (−0.184 − 0.320i)3-s + (−0.366 + 1.96i)4-s + (1.02 − 0.594i)5-s + (−0.181 + 0.490i)6-s + (0.287 − 0.498i)7-s + (2.47 − 1.37i)8-s + (1.43 − 2.47i)9-s + (−1.57 − 0.582i)10-s − 2.93i·11-s + (0.697 − 0.246i)12-s + (−0.418 + 0.725i)13-s + (−0.802 + 0.137i)14-s + (−0.380 − 0.219i)15-s + (−3.73 − 1.44i)16-s + (0.227 − 0.393i)17-s + ⋯
L(s)  = 1  + (−0.638 − 0.769i)2-s + (−0.106 − 0.184i)3-s + (−0.183 + 0.983i)4-s + (0.460 − 0.265i)5-s + (−0.0740 + 0.200i)6-s + (0.108 − 0.188i)7-s + (0.873 − 0.486i)8-s + (0.477 − 0.826i)9-s + (−0.498 − 0.184i)10-s − 0.885i·11-s + (0.201 − 0.0710i)12-s + (−0.116 + 0.201i)13-s + (−0.214 + 0.0367i)14-s + (−0.0983 − 0.0567i)15-s + (−0.932 − 0.360i)16-s + (0.0551 − 0.0954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0162 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0162 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(172\)    =    \(2^{2} \cdot 43\)
Sign: $-0.0162 + 0.999i$
Analytic conductor: \(1.37342\)
Root analytic conductor: \(1.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{172} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 172,\ (\ :1/2),\ -0.0162 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616208 - 0.626323i\)
\(L(\frac12)\) \(\approx\) \(0.616208 - 0.626323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.903 + 1.08i)T \)
43 \( 1 + (-0.537 - 6.53i)T \)
good3 \( 1 + (0.184 + 0.320i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.02 + 0.594i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.287 + 0.498i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.93iT - 11T^{2} \)
13 \( 1 + (0.418 - 0.725i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.227 + 0.393i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.925 + 1.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.24 + 1.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.67 - 2.70i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.87 + 1.66i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.05 - 2.34i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.37T + 41T^{2} \)
47 \( 1 - 7.80iT - 47T^{2} \)
53 \( 1 + (-1.31 - 2.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.617iT - 59T^{2} \)
61 \( 1 + (8.10 + 4.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.58 - 5.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.285 + 0.494i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (12.9 + 7.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.5 - 7.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 - 6.04i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.12 + 4.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.21T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.40490200958640770854117120472, −11.44813506779407563650576580734, −10.51859048182149327340618295811, −9.450759949239175292879623744776, −8.737994972349573154919144978428, −7.45931179626197136726173357852, −6.26804350745795036749689683497, −4.52854932341642424208002287255, −3.05256782881854279292098182205, −1.17743832666115266275631657725, 2.02798256767732519124591251849, 4.55851434262089207950767105773, 5.61373511792155406710702774201, 6.85323221900460260213844137534, 7.77725983141678083081667746763, 8.910732087750454468505498016204, 10.15493069127789158901367637406, 10.42494937979808100621258793561, 11.91736156100371238395532560389, 13.26115876377051402040951872337

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