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Properties

Label 1-55e2-3025.916-r0-0-0
Degree $1$
Conductor $3025$
Sign $-0.651 - 0.758i$
Analytic cond. $14.0480$
Root an. cond. $14.0480$
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  + (−0.654 − 0.755i)2-s + (0.309 − 0.951i)3-s + (−0.142 + 0.989i)4-s + (−0.921 + 0.389i)6-s + (0.941 + 0.336i)7-s + (0.841 − 0.540i)8-s + (−0.809 − 0.587i)9-s + (0.897 + 0.441i)12-s + (0.696 − 0.717i)13-s + (−0.362 − 0.931i)14-s + (−0.959 − 0.281i)16-s + (0.993 − 0.113i)17-s + (0.0855 + 0.996i)18-s + (0.415 − 0.909i)19-s + (0.610 − 0.791i)21-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (0.309 − 0.951i)3-s + (−0.142 + 0.989i)4-s + (−0.921 + 0.389i)6-s + (0.941 + 0.336i)7-s + (0.841 − 0.540i)8-s + (−0.809 − 0.587i)9-s + (0.897 + 0.441i)12-s + (0.696 − 0.717i)13-s + (−0.362 − 0.931i)14-s + (−0.959 − 0.281i)16-s + (0.993 − 0.113i)17-s + (0.0855 + 0.996i)18-s + (0.415 − 0.909i)19-s + (0.610 − 0.791i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.651 - 0.758i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (916, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ -0.651 - 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7160546175 - 1.559361321i\)
\(L(\frac12)\) \(\approx\) \(0.7160546175 - 1.559361321i\)
\(L(1)\) \(\approx\) \(0.8111625010 - 0.6727265789i\)
\(L(1)\) \(\approx\) \(0.8111625010 - 0.6727265789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.941 + 0.336i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.0285 - 0.999i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.696 + 0.717i)T \)
37 \( 1 + (0.696 + 0.717i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.516 + 0.856i)T \)
53 \( 1 + (-0.564 - 0.825i)T \)
59 \( 1 + (0.0855 - 0.996i)T \)
61 \( 1 + (0.0855 - 0.996i)T \)
67 \( 1 + (0.0855 + 0.996i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.564 + 0.825i)T \)
79 \( 1 + (-0.254 + 0.967i)T \)
83 \( 1 + (-0.0285 + 0.999i)T \)
89 \( 1 + (-0.736 + 0.676i)T \)
97 \( 1 + (0.774 + 0.633i)T \)
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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

−19.27943870284540570351565275593, −18.46216360232201196826286340477, −17.83527250553116950269793705221, −17.02740566718113579077182810762, −16.44990770702210081072998709834, −16.00393773521037039218137643739, −15.099228703786704318307188900587, −14.54143882512196089552163309946, −14.04535087437882043559654265477, −13.42078395892868871165385385122, −11.92562234930440908613366892166, −11.274934633171373633335098010305, −10.574561045784720102602948608857, −9.97098241005339710873448578812, −9.2109136782401418354305225561, −8.612820290716492293323586266837, −7.76624069737485081158776960751, −7.44666533131265331683265693464, −6.01884706540740840590339551758, −5.65527922802961864787812037119, −4.63284239150687977868384174359, −4.13043733003338975384952949551, −3.091390415267055020444360735967, −1.80980740386630371319646512264, −1.10200613916287335113072175414, 0.80927714386788096879102082547, 1.21647073219420491714131223914, 2.398636241941797690180514715757, 2.77750696727962745126418487595, 3.77714790017585891083611009345, 4.80658406902635574188670161007, 5.75177357661899482358301712803, 6.70865105743640540943629051554, 7.56821260768061172481282507582, 8.171062534845215020322367095576, 8.57606074704440010743386285316, 9.42598303636157139469698063880, 10.325312749674287613560871987777, 11.1640605127055625942644131208, 11.65327240455104073822586850918, 12.43577199419722432869624864083, 12.897528380049535021559711572798, 13.88397222232874010207521494873, 14.250675797943074777687884168828, 15.35884592983013501283224796797, 16.06570632342683016257613786096, 17.23581730920604523354727296183, 17.56035017548130203924172441529, 18.18304786430946607498023845612, 18.86452132719031080796773128046

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