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Properties

Label 2-171-171.106-c1-0-10
Degree $2$
Conductor $171$
Sign $0.153 + 0.988i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
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  + (−0.269 − 0.466i)2-s + (−1.40 + 1.00i)3-s + (0.854 − 1.48i)4-s − 0.947·5-s + (0.850 + 0.386i)6-s + (1.18 − 2.05i)7-s − 1.99·8-s + (0.970 − 2.83i)9-s + (0.255 + 0.442i)10-s + (1.76 − 3.05i)11-s + (0.286 + 2.94i)12-s + (0.514 − 0.890i)13-s − 1.27·14-s + (1.33 − 0.954i)15-s + (−1.17 − 2.02i)16-s + (−0.347 + 0.602i)17-s + ⋯
L(s)  = 1  + (−0.190 − 0.330i)2-s + (−0.813 + 0.581i)3-s + (0.427 − 0.740i)4-s − 0.423·5-s + (0.347 + 0.157i)6-s + (0.447 − 0.775i)7-s − 0.706·8-s + (0.323 − 0.946i)9-s + (0.0807 + 0.139i)10-s + (0.532 − 0.922i)11-s + (0.0827 + 0.850i)12-s + (0.142 − 0.246i)13-s − 0.341·14-s + (0.344 − 0.246i)15-s + (−0.292 − 0.506i)16-s + (−0.0843 + 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.153 + 0.988i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 171,\ (\ :1/2),\ 0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629148 - 0.538829i\)
\(L(\frac12)\) \(\approx\) \(0.629148 - 0.538829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 - 1.00i)T \)
19 \( 1 + (-2.46 + 3.59i)T \)
good2 \( 1 + (0.269 + 0.466i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.947T + 5T^{2} \)
7 \( 1 + (-1.18 + 2.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.76 + 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.514 + 0.890i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.347 - 0.602i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.69 - 2.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.53T + 29T^{2} \)
31 \( 1 + (-4.48 - 7.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.345T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + (-2.10 - 3.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + (3.33 + 5.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 + (-1.02 + 1.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.75 + 3.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.57 - 7.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.87 + 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.41 + 4.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.902 + 1.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.02 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.94628468352454163266976065638, −11.45287359386086060205713736468, −10.64715455952400671916689433365, −9.894256782774526488363309278868, −8.668740153634360923302410806791, −7.09673019379811315254227011355, −6.05300644117724004588088302476, −4.90090415174835357171875596658, −3.50147174704844096912850192189, −0.968155592316979274910006269918, 2.17141747747639859339438519442, 4.20856716448416238275068385277, 5.73704032729122835262940721343, 6.79233139697472645321020916143, 7.70894092315960960116898958161, 8.561767781060179800358772967790, 10.03457604699364588548634358111, 11.54781768200174527921297078986, 11.89428233876477100662521717523, 12.55557318073402515130093161985

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