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Properties

Label 2-605-5.4-c1-0-27
Degree $2$
Conductor $605$
Sign $1$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.32i·2-s + 0.236·4-s − 2.23·5-s − 4.29i·7-s + 2.96i·8-s + 3·9-s − 2.96i·10-s − 6.95i·13-s + 5.70·14-s − 3.47·16-s − 1.64i·17-s + 3.98i·18-s − 0.527·20-s + 5.00·25-s + 9.23·26-s + ⋯
L(s)  = 1  + 0.939i·2-s + 0.118·4-s − 0.999·5-s − 1.62i·7-s + 1.04i·8-s + 9-s − 0.939i·10-s − 1.92i·13-s + 1.52·14-s − 0.868·16-s − 0.398i·17-s + 0.939i·18-s − 0.118·20-s + 1.00·25-s + 1.81·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42620\)
\(L(\frac12)\) \(\approx\) \(1.42620\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2.23T \)
11 \( 1 \)
good2 \( 1 - 1.32iT - 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 + 4.29iT - 7T^{2} \)
13 \( 1 + 6.95iT - 13T^{2} \)
17 \( 1 + 1.64iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 18.2iT - 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.56773429809050702094125430593, −10.02158095738720448841040376047, −8.383725431406557495641381701981, −7.69973680062855230818673182622, −7.29148769442485065294208979565, −6.45618254281969152852327513068, −5.09364302244767252011136695758, −4.21503985637338129807853813363, −3.08236766931578157297296305224, −0.859393174673052303896204801828, 1.61141406837569076504724973586, 2.65372624563422364466919908071, 3.90201486553940857797504435134, 4.70626903179474012636819906792, 6.32188019999766908839703632204, 6.98047121238297238110909992043, 8.177517812391450340447616275757, 9.125620286717069353757234417276, 9.789608227113462546057139821718, 10.92435672948585458279956915262

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