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Properties

Label 1-1205-1205.112-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.955 - 0.294i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.258 − 0.965i)2-s + (0.544 − 0.838i)3-s + (−0.866 + 0.5i)4-s + (−0.951 − 0.309i)6-s + (−0.878 + 0.477i)7-s + (0.707 + 0.707i)8-s + (−0.406 − 0.913i)9-s + (−0.130 + 0.991i)11-s + (−0.0523 + 0.998i)12-s + (0.182 − 0.983i)13-s + (0.688 + 0.725i)14-s + (0.5 − 0.866i)16-s + (0.233 + 0.972i)17-s + (−0.777 + 0.629i)18-s + (0.991 + 0.130i)19-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.544 − 0.838i)3-s + (−0.866 + 0.5i)4-s + (−0.951 − 0.309i)6-s + (−0.878 + 0.477i)7-s + (0.707 + 0.707i)8-s + (−0.406 − 0.913i)9-s + (−0.130 + 0.991i)11-s + (−0.0523 + 0.998i)12-s + (0.182 − 0.983i)13-s + (0.688 + 0.725i)14-s + (0.5 − 0.866i)16-s + (0.233 + 0.972i)17-s + (−0.777 + 0.629i)18-s + (0.991 + 0.130i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.955 - 0.294i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.955 - 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1551031094 - 1.028619694i\)
\(L(\frac12)\) \(\approx\) \(0.1551031094 - 1.028619694i\)
\(L(1)\) \(\approx\) \(0.6628220402 - 0.6010713596i\)
\(L(1)\) \(\approx\) \(0.6628220402 - 0.6010713596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.544 - 0.838i)T \)
7 \( 1 + (-0.878 + 0.477i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (0.182 - 0.983i)T \)
17 \( 1 + (0.233 + 0.972i)T \)
19 \( 1 + (0.991 + 0.130i)T \)
23 \( 1 + (-0.996 + 0.0784i)T \)
29 \( 1 + (0.358 - 0.933i)T \)
31 \( 1 + (0.284 - 0.958i)T \)
37 \( 1 + (-0.182 - 0.983i)T \)
41 \( 1 + (0.987 + 0.156i)T \)
43 \( 1 + (0.522 + 0.852i)T \)
47 \( 1 + (-0.156 - 0.987i)T \)
53 \( 1 + (-0.629 - 0.777i)T \)
59 \( 1 + (-0.544 - 0.838i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (-0.777 - 0.629i)T \)
71 \( 1 + (-0.999 + 0.0261i)T \)
73 \( 1 + (0.760 - 0.649i)T \)
79 \( 1 + (0.453 - 0.891i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.793 - 0.608i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−21.784549703572948790462396937948, −20.6920997537427900325629162708, −19.88488872245067366626200694949, −19.14484502474884928270701620496, −18.548550851109852551427951766707, −17.47876117593324432038464678415, −16.45425028855915503088617677065, −16.111685896468631637024333235, −15.77344320418367282145453498349, −14.477308491640784741317340365592, −13.7865497716133162530615615298, −13.68223213950630592901874376453, −12.236307460733805831644989833290, −11.009145381473066005468010778, −10.24570134742880144084310124265, −9.420135534461703327385489088521, −8.996440713128669408549516910083, −8.026928196705030395012982116591, −7.1971372975369166334830559647, −6.3114697447571920462310633405, −5.40261891126510555449215514643, −4.51740838180605268785416466903, −3.63928729938961094546699625223, −2.85489485408033814744315653635, −1.10251236898678523876997438264, 0.49025624190437209551910999744, 1.73773242637610312104409277876, 2.49416102931426111682488071320, 3.30674599741680142009165856022, 4.11109449608911741908217544036, 5.537611736639507716076546338757, 6.33017328649032375278903040706, 7.702931539712472577003313810877, 7.96059695601737764263163535224, 9.13547376790174155412776051371, 9.73156518687028073877686057523, 10.43692324216032800752809254574, 11.72857749254536048570640777782, 12.295591173561742394093282465465, 12.92909105036592106832265471498, 13.45098735670768393695107491463, 14.47364671920325572234221669634, 15.275299191031933481723911880736, 16.262156540732078374102243840827, 17.56381506433852247168645925983, 17.84463891783408336751199450293, 18.67902766582749066240433376326, 19.43114403895969374523713156094, 19.9015878714590795215574971730, 20.605471256408777637129601066670

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