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Properties

Label 1-55e2-3025.2754-r0-0-0
Degree $1$
Conductor $3025$
Sign $0.724 + 0.689i$
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  + (0.0285 − 0.999i)2-s + (−0.309 + 0.951i)3-s + (−0.998 − 0.0570i)4-s + (0.941 + 0.336i)6-s + (0.870 + 0.491i)7-s + (−0.0855 + 0.996i)8-s + (−0.809 − 0.587i)9-s + (0.362 − 0.931i)12-s + (0.998 + 0.0570i)13-s + (0.516 − 0.856i)14-s + (0.993 + 0.113i)16-s + (−0.696 + 0.717i)17-s + (−0.610 + 0.791i)18-s + (0.696 + 0.717i)19-s + (−0.736 + 0.676i)21-s + ⋯
L(s)  = 1  + (0.0285 − 0.999i)2-s + (−0.309 + 0.951i)3-s + (−0.998 − 0.0570i)4-s + (0.941 + 0.336i)6-s + (0.870 + 0.491i)7-s + (−0.0855 + 0.996i)8-s + (−0.809 − 0.587i)9-s + (0.362 − 0.931i)12-s + (0.998 + 0.0570i)13-s + (0.516 − 0.856i)14-s + (0.993 + 0.113i)16-s + (−0.696 + 0.717i)17-s + (−0.610 + 0.791i)18-s + (0.696 + 0.717i)19-s + (−0.736 + 0.676i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ 0.724 + 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400877773 + 0.5597018078i\)
\(L(\frac12)\) \(\approx\) \(1.400877773 + 0.5597018078i\)
\(L(1)\) \(\approx\) \(1.016851136 + 0.005389854320i\)
\(L(1)\) \(\approx\) \(1.016851136 + 0.005389854320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0285 - 0.999i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.870 + 0.491i)T \)
13 \( 1 + (0.998 + 0.0570i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (0.870 - 0.491i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.774 + 0.633i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.564 - 0.825i)T \)
53 \( 1 + (0.870 + 0.491i)T \)
59 \( 1 + (-0.564 + 0.825i)T \)
61 \( 1 + (0.941 - 0.336i)T \)
67 \( 1 + (0.0285 - 0.999i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.974 + 0.226i)T \)
83 \( 1 + (-0.198 + 0.980i)T \)
89 \( 1 + (0.696 - 0.717i)T \)
97 \( 1 + (-0.0855 + 0.996i)T \)
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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

−18.7819354796061676494726272896, −17.98456450959405296755560112322, −17.56681386461437517496383524931, −17.11674417265107327102485411804, −16.1516143814647491724248706152, −15.5761744766564899181289352594, −14.71858851970993726802379030023, −13.89206765984611727938818812604, −13.484543200535867744909964997213, −13.02814017350838845900979848531, −11.83302074954652642665378644228, −11.332701947244230677695398298276, −10.55374785812153822667099708921, −9.37378742917389171615347593085, −8.69009315705379881986511208085, −7.983502289792009588491743754689, −7.32041937582173305337381770419, −6.82718002615464512458853175319, −5.961075469684756610902833490209, −5.24771503954162599243175648814, −4.58541399621224635100632904517, −3.608654208843440997316175449618, −2.46773182891360341293776943088, −1.283406118168690589693014719565, −0.610241076928878274538563453803, 1.03022812256756361862895809026, 1.84764788410841941656583778012, 2.91121216571029693554591811582, 3.65154571018803134787206137120, 4.341623714211843901692415974592, 5.15568300861750617457959113605, 5.622378221843156837967186654728, 6.651782115740334084349410072316, 8.07123837778970503174464734668, 8.7546445000165870503828781568, 9.04038331076136958828730746757, 10.20150084319970438987928817178, 10.65153155294152878257058296935, 11.23948523679722888343647026861, 11.92771817877109543325738701203, 12.493484926876091804043691897666, 13.60190165582343027732839608004, 14.11218720654654056689481537537, 15.0516839449192289407164818981, 15.389125016851650169635651789094, 16.49192324754730524131273834315, 17.13818860344097013603206565464, 17.87388190878675823027167240149, 18.41807511027989872780188456402, 19.11514859712075992424391330332

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