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Properties

Label 2-1205-5.4-c1-0-80
Degree $2$
Conductor $1205$
Sign $0.964 - 0.265i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
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  − 0.185i·2-s + 0.636i·3-s + 1.96·4-s + (2.15 − 0.593i)5-s + 0.118·6-s + 3.56i·7-s − 0.735i·8-s + 2.59·9-s + (−0.110 − 0.399i)10-s + 1.87·11-s + 1.25i·12-s − 2.26i·13-s + 0.661·14-s + (0.377 + 1.37i)15-s + 3.79·16-s − 7.09i·17-s + ⋯
L(s)  = 1  − 0.131i·2-s + 0.367i·3-s + 0.982·4-s + (0.964 − 0.265i)5-s + 0.0481·6-s + 1.34i·7-s − 0.259i·8-s + 0.864·9-s + (−0.0348 − 0.126i)10-s + 0.564·11-s + 0.361i·12-s − 0.628i·13-s + 0.176·14-s + (0.0975 + 0.354i)15-s + 0.948·16-s − 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.703073906\)
\(L(\frac12)\) \(\approx\) \(2.703073906\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.15 + 0.593i)T \)
241 \( 1 - T \)
good2 \( 1 + 0.185iT - 2T^{2} \)
3 \( 1 - 0.636iT - 3T^{2} \)
7 \( 1 - 3.56iT - 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 2.26iT - 13T^{2} \)
17 \( 1 + 7.09iT - 17T^{2} \)
19 \( 1 + 3.44T + 19T^{2} \)
23 \( 1 - 2.84iT - 23T^{2} \)
29 \( 1 + 1.58T + 29T^{2} \)
31 \( 1 + 3.99T + 31T^{2} \)
37 \( 1 - 1.44iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 8.31iT - 43T^{2} \)
47 \( 1 + 1.88iT - 47T^{2} \)
53 \( 1 + 7.12iT - 53T^{2} \)
59 \( 1 + 7.62T + 59T^{2} \)
61 \( 1 + 5.88T + 61T^{2} \)
67 \( 1 - 0.787iT - 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 1.01iT - 73T^{2} \)
79 \( 1 + 8.92T + 79T^{2} \)
83 \( 1 - 5.72iT - 83T^{2} \)
89 \( 1 - 4.93T + 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−9.639902677739720815813230491044, −9.295841046142515610781031829626, −8.247664313853467151427549452997, −7.11988048569208952110254995854, −6.41555012098286004123766297298, −5.53744580237366880029811917196, −4.87580598150240481789011469892, −3.34669978161310582763817084838, −2.40381269004787875658160191826, −1.50637746505265663267551085256, 1.46809383026138643168667677378, 1.97616724209849848919188083387, 3.56362811558790871160956102723, 4.40259751139267594233947174388, 5.86398174705620673220911841225, 6.63548869285369525522836974145, 6.93870263274069798138237828723, 7.82249648382706286020785615145, 8.875800748185043086421735242936, 9.998493217719403067969889450977

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