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Properties

Label 1-180-180.103-r0-0-0
Degree $1$
Conductor $180$
Sign $0.619 - 0.784i$
Analytic cond. $0.835916$
Root an. cond. $0.835916$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s − i·17-s + 19-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s − i·37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s + (0.5 + 0.866i)49-s + i·53-s + (−0.5 − 0.866i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s − i·17-s + 19-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s − i·37-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s + (0.5 + 0.866i)49-s + i·53-s + (−0.5 − 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.619 - 0.784i$
Analytic conductor: \(0.835916\)
Root analytic conductor: \(0.835916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 180,\ (0:\ ),\ 0.619 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9495318916 - 0.4602231491i\)
\(L(\frac12)\) \(\approx\) \(0.9495318916 - 0.4602231491i\)
\(L(1)\) \(\approx\) \(0.9932061050 - 0.2006857908i\)
\(L(1)\) \(\approx\) \(0.9932061050 - 0.2006857908i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 + T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.66857156291805264024609153752, −26.19821677793104865592344914207, −25.7422062627983920615409643532, −24.662894084338377581195659487087, −23.59902782394974141978439881327, −22.56382180121813583667587860308, −21.89091841068258405995197363219, −20.63799047934211511777389455934, −19.72864575300969468685817053808, −18.74628050463900670532404041834, −17.82406890398442171492886473654, −16.59894760737686537576120890115, −15.75266391419677700459798655526, −14.73242064784282077055427454335, −13.55410121485110522875064761202, −12.509286126567353094034239841438, −11.64758204806397681604212559287, −10.22362519358324364568175750206, −9.342582437927027999059610857336, −8.23162708800077380170417330891, −6.76766448682723434216717310651, −5.95944605187196083986686695306, −4.40214035112962352645537886000, −3.20159152874674930796969735462, −1.66565726402472241815361418806, 0.938233381547642168904106781351, 2.975344835202303108590410400202, 3.9095395532430418668307301454, 5.56210090651957534747492329506, 6.564411785746478074097098740411, 7.76623829328957830807320725434, 9.03803818027556563307862346944, 10.02937177502356290641910606957, 11.16832162846060285637782555208, 12.222292661173538327023009411992, 13.570771950784608454571089218743, 14.007658176734994785593692744351, 15.7829784462555658009346478554, 16.17482144336132319399123562679, 17.453156410996926075463308551530, 18.48387437286542438942711141644, 19.53184149208388347242844865690, 20.31105519360159571983256243395, 21.45151319133627261246892573044, 22.56239738322212448812516435936, 23.16553773004253715806186175541, 24.423537746949986519351368212937, 25.2626295545272811545971103270, 26.31311123283726537661806409912, 27.06822445086163963938830700733

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