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Properties

Label 2-172-172.7-c1-0-17
Degree $2$
Conductor $172$
Sign $0.0205 + 0.999i$
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  + (1.36 − 0.370i)2-s + (−1.34 − 2.33i)3-s + (1.72 − 1.01i)4-s + (1.44 − 0.831i)5-s + (−2.69 − 2.68i)6-s + (−2.17 + 3.76i)7-s + (1.98 − 2.01i)8-s + (−2.12 + 3.67i)9-s + (1.65 − 1.66i)10-s − 3.65i·11-s + (−4.67 − 2.66i)12-s + (0.128 − 0.223i)13-s + (−1.57 + 5.93i)14-s + (−3.87 − 2.23i)15-s + (1.95 − 3.48i)16-s + (−1.34 + 2.32i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.261i)2-s + (−0.776 − 1.34i)3-s + (0.863 − 0.505i)4-s + (0.644 − 0.372i)5-s + (−1.10 − 1.09i)6-s + (−0.820 + 1.42i)7-s + (0.700 − 0.713i)8-s + (−0.707 + 1.22i)9-s + (0.524 − 0.527i)10-s − 1.10i·11-s + (−1.35 − 0.769i)12-s + (0.0357 − 0.0619i)13-s + (−0.420 + 1.58i)14-s + (−1.00 − 0.578i)15-s + (0.489 − 0.871i)16-s + (−0.325 + 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0205 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0205 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(172\)    =    \(2^{2} \cdot 43\)
Sign: $0.0205 + 0.999i$
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: \((2,\ 172,\ (\ :1/2),\ 0.0205 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16491 - 1.14123i\)
\(L(\frac12)\) \(\approx\) \(1.16491 - 1.14123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.370i)T \)
43 \( 1 + (5.42 + 3.67i)T \)
good3 \( 1 + (1.34 + 2.33i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.44 + 0.831i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.17 - 3.76i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.65iT - 11T^{2} \)
13 \( 1 + (-0.128 + 0.223i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.34 - 2.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 - 7.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.33 + 2.50i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.59 - 0.923i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.08 + 1.20i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.668 + 0.386i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.12T + 41T^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 + (2.32 + 4.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.456iT - 59T^{2} \)
61 \( 1 + (7.02 + 4.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.84 - 5.68i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.85 + 8.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.65 - 4.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.48 - 1.43i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.05 - 5.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.16 - 0.671i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−12.49787659839224804519250709980, −11.99328930258630574058197238605, −10.99820225886474236069746464907, −9.620676392508159558171224057859, −8.225046007779713774579952250676, −6.66700502270823548808904386227, −5.90258919944934585981332452180, −5.45150609799323403835770035860, −3.07657759965807851202755105310, −1.59622974992471849911114832773, 3.12869916701633603081017833614, 4.39833808231900020269115959877, 5.12971104629747615172775993788, 6.57231636836530470305689244169, 7.19536025777965071808965873023, 9.480999788050628909615071915305, 10.17474358818107724404335908635, 10.95993053241359801845226042519, 11.89354992461795208891868201005, 13.33170951913910736807792164709

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