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Properties

Label 1-40e2-1600.989-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.726 + 0.687i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (0.522 + 0.852i)3-s + (−0.707 + 0.707i)7-s + (−0.453 + 0.891i)9-s + (0.0784 + 0.996i)11-s + (0.996 + 0.0784i)13-s + (0.587 + 0.809i)17-s + (0.233 + 0.972i)19-s + (−0.972 − 0.233i)21-s + (0.891 − 0.453i)23-s + (−0.996 + 0.0784i)27-s + (0.522 + 0.852i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)33-s + (0.760 − 0.649i)37-s + (0.453 + 0.891i)39-s + ⋯
L(s)  = 1  + (0.522 + 0.852i)3-s + (−0.707 + 0.707i)7-s + (−0.453 + 0.891i)9-s + (0.0784 + 0.996i)11-s + (0.996 + 0.0784i)13-s + (0.587 + 0.809i)17-s + (0.233 + 0.972i)19-s + (−0.972 − 0.233i)21-s + (0.891 − 0.453i)23-s + (−0.996 + 0.0784i)27-s + (0.522 + 0.852i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)33-s + (0.760 − 0.649i)37-s + (0.453 + 0.891i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.726 + 0.687i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.726 + 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6697063455 + 1.681829599i\)
\(L(\frac12)\) \(\approx\) \(0.6697063455 + 1.681829599i\)
\(L(1)\) \(\approx\) \(1.051382660 + 0.6827257076i\)
\(L(1)\) \(\approx\) \(1.051382660 + 0.6827257076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.522 + 0.852i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.0784 + 0.996i)T \)
13 \( 1 + (0.996 + 0.0784i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.233 + 0.972i)T \)
23 \( 1 + (0.891 - 0.453i)T \)
29 \( 1 + (0.522 + 0.852i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.760 - 0.649i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.972 - 0.233i)T \)
59 \( 1 + (0.760 - 0.649i)T \)
61 \( 1 + (0.649 - 0.760i)T \)
67 \( 1 + (0.972 - 0.233i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (-0.233 - 0.972i)T \)
89 \( 1 + (0.453 + 0.891i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−20.08149863864969224193578366412, −19.335100370750105432594013312129, −18.89412763531798795766196833060, −18.06450908354965869527717379765, −17.31179812576229009314930840392, −16.42397540010584304499846327922, −15.79501568966430858070707641460, −14.83760152506077753586393670705, −13.83032129324839604954991180903, −13.47187991259853037494959121374, −12.995542546167728364248983990732, −11.75096465718852024757768934412, −11.319403372396110884309700154206, −10.188306915569347636371247562450, −9.35547233228667089444298664735, −8.57730562595593039273388130735, −7.86165012761338348690472056987, −6.91737409342981934174565510965, −6.42981714217556464068909811521, −5.502857216445586512263826798570, −4.23718140392797251246391664141, −3.04374548055379164173099885223, −2.99833220937316173010277344962, −1.276169620096226226862564401024, −0.69330647889755115712311343510, 1.46148625455834165052641183353, 2.508759859700631472049741294939, 3.36607208450019277833572754362, 4.055025856422322161853352639738, 5.038409193519988235686392931775, 5.886393723554268069096697569611, 6.69835906702214809223751523628, 7.91833339217922524046391557462, 8.53874060012111629913555079956, 9.387425262749392689628758019223, 9.954063972399179991891121350, 10.69047806055560802339946470777, 11.62354269960334769310135170404, 12.62704186273257941865009714924, 13.09122785830283592471684701792, 14.33837583294348339175064466572, 14.70990548661957067061650034701, 15.62246662154976770515544365418, 16.08565623429714931816337363737, 16.849322181097950528135976323804, 17.76088945726353736518684556445, 18.79926097726740058146506762052, 19.18529011637634999494254358689, 20.18199060746149050490068582744, 20.75962649298113295881140615450

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