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Properties

Label 1-55e2-3025.841-r0-0-0
Degree $1$
Conductor $3025$
Sign $0.00675 - 0.999i$
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  + (−0.998 − 0.0570i)2-s + (−0.809 − 0.587i)3-s + (0.993 + 0.113i)4-s + (0.774 + 0.633i)6-s + (0.516 + 0.856i)7-s + (−0.985 − 0.170i)8-s + (0.309 + 0.951i)9-s + (−0.736 − 0.676i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)14-s + (0.974 + 0.226i)16-s + (−0.0285 − 0.999i)17-s + (−0.254 − 0.967i)18-s + (−0.0285 + 0.999i)19-s + (0.0855 − 0.996i)21-s + ⋯
L(s)  = 1  + (−0.998 − 0.0570i)2-s + (−0.809 − 0.587i)3-s + (0.993 + 0.113i)4-s + (0.774 + 0.633i)6-s + (0.516 + 0.856i)7-s + (−0.985 − 0.170i)8-s + (0.309 + 0.951i)9-s + (−0.736 − 0.676i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)14-s + (0.974 + 0.226i)16-s + (−0.0285 − 0.999i)17-s + (−0.254 − 0.967i)18-s + (−0.0285 + 0.999i)19-s + (0.0855 − 0.996i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00675 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00675 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.00675 - 0.999i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ 0.00675 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5153587615 - 0.5118915067i\)
\(L(\frac12)\) \(\approx\) \(0.5153587615 - 0.5118915067i\)
\(L(1)\) \(\approx\) \(0.5901634033 - 0.1239784065i\)
\(L(1)\) \(\approx\) \(0.5901634033 - 0.1239784065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 - 0.0570i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.516 + 0.856i)T \)
13 \( 1 + (0.993 + 0.113i)T \)
17 \( 1 + (-0.0285 - 0.999i)T \)
19 \( 1 + (-0.0285 + 0.999i)T \)
23 \( 1 + (0.516 - 0.856i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.362 - 0.931i)T \)
53 \( 1 + (0.516 + 0.856i)T \)
59 \( 1 + (-0.362 - 0.931i)T \)
61 \( 1 + (0.774 - 0.633i)T \)
67 \( 1 + (-0.998 - 0.0570i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (0.897 + 0.441i)T \)
83 \( 1 + (-0.921 - 0.389i)T \)
89 \( 1 + (-0.0285 - 0.999i)T \)
97 \( 1 + (-0.985 - 0.170i)T \)
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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

−19.216541208044503047334168385553, −18.21476472471908283824960412620, −17.71463086312279937852818240634, −17.24609590492844862369187431495, −16.504409139428611029548242140453, −16.05495970626056173855136022937, −15.04764943577889600949351625128, −14.85111102740238720111291866274, −13.466798958787623261642715325038, −12.846238292992910244235330217081, −11.66778245393842301063222106753, −11.20223764552017849772449467430, −10.77264979957575737163508292038, −10.042070453916242253088433884225, −9.32770137371172017419847312596, −8.55594327415737531912888356582, −7.76049780899130225835440026239, −6.96205271205333549024005465032, −6.26139842472617216703305009925, −5.56423806924002603042356365602, −4.56987982261419520892744042563, −3.78405609462250441347508149685, −2.89029353553923428079565031954, −1.42126580574646087018317082679, −1.02704562711365198665283634867, 0.41810909562916107755846873564, 1.43323703093770413660248775849, 2.09204730587504155194800125217, 2.947041545244421313958139321283, 4.21018119953720451332235628301, 5.38669805628939167895895051543, 5.85986765208544883797802452145, 6.65820119636200911432422151432, 7.37747311392354894405805529431, 8.1658166830854866026679912081, 8.741777533021628183510190052402, 9.54164873593054767261320434659, 10.57048087698811373680268382442, 10.98151801149084359311789941705, 11.83897224307701088900096124333, 12.14193045135324575441416437879, 13.00304082541793412318017583948, 13.97423164592223980263575289482, 14.81869337116369207187305572467, 15.78390054074865326564956226086, 16.166892613803515551645275835695, 16.92426467165965947195784961654, 17.66702624413801720667276367263, 18.22665685453931647002864544634, 18.73054119536780114183967736768

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