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Properties

Label 1-2475-2475.1447-r0-0-0
Degree $1$
Conductor $2475$
Sign $-0.915 - 0.401i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
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)2-s + (0.5 − 0.866i)4-s + (0.207 + 0.978i)7-s + i·8-s + (−0.743 + 0.669i)13-s + (−0.669 − 0.743i)14-s + (−0.5 − 0.866i)16-s + (0.587 + 0.809i)17-s + 19-s + (−0.406 − 0.913i)23-s + (0.309 − 0.951i)26-s + (0.951 + 0.309i)28-s + (−0.5 − 0.866i)29-s + (−0.978 − 0.207i)31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.207 + 0.978i)7-s + i·8-s + (−0.743 + 0.669i)13-s + (−0.669 − 0.743i)14-s + (−0.5 − 0.866i)16-s + (0.587 + 0.809i)17-s + 19-s + (−0.406 − 0.913i)23-s + (0.309 − 0.951i)26-s + (0.951 + 0.309i)28-s + (−0.5 − 0.866i)29-s + (−0.978 − 0.207i)31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.915 - 0.401i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ -0.915 - 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06856312970 + 0.3273413224i\)
\(L(\frac12)\) \(\approx\) \(-0.06856312970 + 0.3273413224i\)
\(L(1)\) \(\approx\) \(0.5695926262 + 0.2446551538i\)
\(L(1)\) \(\approx\) \(0.5695926262 + 0.2446551538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (-0.743 + 0.669i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.406 - 0.913i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.994 - 0.104i)T \)
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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

−19.10840161662362833495938589504, −18.29178163797471585730708926863, −17.68161716773235881515964193083, −17.15125911625653450362068443134, −16.301863517361471505276995377813, −15.87656393333495418806353367623, −14.760041991646604769981588737739, −14.00115205980007716742250399202, −13.21025691610186828093918409234, −12.43581153154283741996589484437, −11.68869774230524583848565555521, −11.04690566910532989204658946116, −10.20170296329961540208593513247, −9.771158625723178596451201879343, −8.967954253922293158514730593842, −7.96237766456975629102196480235, −7.35201164724594753463409681983, −7.00151038697387496896188897147, −5.58522105023423282563164661029, −4.835161516947738618619917335, −3.545298848637591701228817801229, −3.26304949280357440696002555257, −1.99773563638268161004539574865, −1.19293939762225555539168942874, −0.14903322881343706893779686873, 1.38666901607050258697480838391, 2.11008680836750274489039468005, 2.97130978144394623766832887939, 4.28864267895377845833319195254, 5.277800024645424942750002569274, 5.83103777556368509975874246376, 6.64904520907760869264183258496, 7.530869014681483265854519197369, 8.14047802200989734555656689878, 8.96772557960885139964922087658, 9.52793946260836781662572999419, 10.25518889427855406826918452412, 11.12099349360129534747155939558, 11.93394108085649361931708166822, 12.38662698057760230395873452342, 13.65030650102383201559962045272, 14.49089701323121161715591627863, 14.92340020674672169470456017724, 15.68722690639844993575065947871, 16.4235920090441695413826458904, 17.00914160990456002369241044437, 17.78982113862650182687238665187, 18.46522772455622938561893590836, 19.0246413332997781192083394806, 19.57489896591446817959005838336

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