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Properties

Label 2-172-172.123-c1-0-0
Degree $2$
Conductor $172$
Sign $-0.974 + 0.225i$
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  + (0.494 + 1.32i)2-s + (−0.862 + 1.49i)3-s + (−1.51 + 1.31i)4-s + (−3.22 − 1.86i)5-s + (−2.40 − 0.403i)6-s + (0.0286 + 0.0496i)7-s + (−2.48 − 1.35i)8-s + (0.0138 + 0.0240i)9-s + (0.872 − 5.19i)10-s + 6.50i·11-s + (−0.653 − 3.38i)12-s + (−1.17 − 2.04i)13-s + (−0.0516 + 0.0625i)14-s + (5.56 − 3.21i)15-s + (0.566 − 3.95i)16-s + (1.70 + 2.95i)17-s + ⋯
L(s)  = 1  + (0.349 + 0.936i)2-s + (−0.497 + 0.862i)3-s + (−0.755 + 0.655i)4-s + (−1.44 − 0.832i)5-s + (−0.981 − 0.164i)6-s + (0.0108 + 0.0187i)7-s + (−0.877 − 0.478i)8-s + (0.00463 + 0.00802i)9-s + (0.275 − 1.64i)10-s + 1.96i·11-s + (−0.188 − 0.977i)12-s + (−0.326 − 0.565i)13-s + (−0.0138 + 0.0167i)14-s + (1.43 − 0.828i)15-s + (0.141 − 0.989i)16-s + (0.413 + 0.715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(172\)    =    \(2^{2} \cdot 43\)
Sign: $-0.974 + 0.225i$
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: \((2,\ 172,\ (\ :1/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0719392 - 0.630468i\)
\(L(\frac12)\) \(\approx\) \(0.0719392 - 0.630468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.494 - 1.32i)T \)
43 \( 1 + (-2.55 + 6.03i)T \)
good3 \( 1 + (0.862 - 1.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.22 + 1.86i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.0286 - 0.0496i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 6.50iT - 11T^{2} \)
13 \( 1 + (1.17 + 2.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.70 - 2.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.342 - 0.594i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.84 - 1.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.67 - 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.93 + 2.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.37 - 4.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
47 \( 1 - 8.53iT - 47T^{2} \)
53 \( 1 + (-2.84 + 4.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.15iT - 59T^{2} \)
61 \( 1 + (3.73 - 2.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.47 - 2.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.38 + 2.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.1 + 5.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.2 - 5.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.33 + 3.65i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.92 + 3.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.61T + 97T^{2} \)
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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

−12.95770549126751440030695915525, −12.50865397707612058051672716002, −11.52084222484972328749485117947, −10.14131170350747695066581548450, −9.147086959192556359558952090878, −7.85135719832788162157172650747, −7.29271831150317824779734511218, −5.43266505330579465695414675379, −4.59382709911366118610983328804, −3.90250213276284591881714225893, 0.57552595859382496521970233980, 2.96075448303303146046682907769, 4.02128453254812293006559341988, 5.74752304004489666204961838015, 6.89825295315414524757947621171, 8.003696475448299609456117476570, 9.266405817456132820561883729443, 10.94368219723189708512419966205, 11.29095494051524881670930329528, 12.00159486315449185562909830989

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