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Properties

Label 2-172-172.123-c1-0-6
Degree $2$
Conductor $172$
Sign $0.794 + 0.607i$
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  + (−0.958 − 1.03i)2-s + (0.555 − 0.961i)3-s + (−0.160 + 1.99i)4-s + (2.85 + 1.64i)5-s + (−1.53 + 0.345i)6-s + (0.143 + 0.249i)7-s + (2.22 − 1.74i)8-s + (0.883 + 1.53i)9-s + (−1.02 − 4.54i)10-s + 2.52i·11-s + (1.82 + 1.26i)12-s + (−0.996 − 1.72i)13-s + (0.121 − 0.388i)14-s + (3.16 − 1.82i)15-s + (−3.94 − 0.640i)16-s + (−2.42 − 4.19i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.734i)2-s + (0.320 − 0.555i)3-s + (−0.0803 + 0.996i)4-s + (1.27 + 0.736i)5-s + (−0.625 + 0.140i)6-s + (0.0543 + 0.0941i)7-s + (0.787 − 0.616i)8-s + (0.294 + 0.510i)9-s + (−0.323 − 1.43i)10-s + 0.760i·11-s + (0.527 + 0.364i)12-s + (−0.276 − 0.478i)13-s + (0.0323 − 0.103i)14-s + (0.817 − 0.471i)15-s + (−0.987 − 0.160i)16-s + (−0.588 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(172\)    =    \(2^{2} \cdot 43\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(1.37342\)
Root analytic conductor: \(1.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{172} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 172,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03561 - 0.350444i\)
\(L(\frac12)\) \(\approx\) \(1.03561 - 0.350444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.958 + 1.03i)T \)
43 \( 1 + (5.79 + 3.06i)T \)
good3 \( 1 + (-0.555 + 0.961i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.85 - 1.64i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.143 - 0.249i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 + (0.996 + 1.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.42 + 4.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.46 + 4.26i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.32 + 3.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.96 - 4.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.88 - 3.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.08 - 0.625i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
47 \( 1 - 3.68iT - 47T^{2} \)
53 \( 1 + (-0.325 + 0.562i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.94iT - 59T^{2} \)
61 \( 1 + (-2.74 + 1.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 + 6.11i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.63 - 4.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.265 + 0.153i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.80 - 3.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.689 - 0.398i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.93 + 5.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−12.68978132131089308307906553186, −11.56765078298448979187033765460, −10.36590955448726568964002115126, −9.878999634164551015814338966375, −8.757391046400062618617698041299, −7.45523084838837345870266529069, −6.72804289169507625415217025881, −4.90509072427209040323927282326, −2.78207429620137287829342001960, −1.94100669784120001382328248330, 1.68322819691783400547216425060, 4.16776186039988336168415032875, 5.63297543138971133656241296903, 6.32606644710269663026346777350, 7.963110741287484881152371840805, 8.934485727842348009413898221412, 9.714803086247019335513422447953, 10.23868379846312208621508667036, 11.73935797398482961820414195324, 13.25118589843313518598565239107

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