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Properties

Label 2-1205-5.4-c1-0-15
Degree $2$
Conductor $1205$
Sign $-0.885 + 0.464i$
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  − 0.257i·2-s + 3.28i·3-s + 1.93·4-s + (−1.98 + 1.03i)5-s + 0.843·6-s + 3.60i·7-s − 1.01i·8-s − 7.77·9-s + (0.267 + 0.509i)10-s − 1.43·11-s + 6.34i·12-s − 2.88i·13-s + 0.926·14-s + (−3.40 − 6.49i)15-s + 3.60·16-s + 0.875i·17-s + ⋯
L(s)  = 1  − 0.181i·2-s + 1.89i·3-s + 0.966·4-s + (−0.885 + 0.464i)5-s + 0.344·6-s + 1.36i·7-s − 0.357i·8-s − 2.59·9-s + (0.0844 + 0.160i)10-s − 0.433·11-s + 1.83i·12-s − 0.799i·13-s + 0.247·14-s + (−0.880 − 1.67i)15-s + 0.901·16-s + 0.212i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.885 + 0.464i$
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: \((2,\ 1205,\ (\ :1/2),\ -0.885 + 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.137091840\)
\(L(\frac12)\) \(\approx\) \(1.137091840\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.98 - 1.03i)T \)
241 \( 1 - T \)
good2 \( 1 + 0.257iT - 2T^{2} \)
3 \( 1 - 3.28iT - 3T^{2} \)
7 \( 1 - 3.60iT - 7T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
13 \( 1 + 2.88iT - 13T^{2} \)
17 \( 1 - 0.875iT - 17T^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 - 4.70iT - 23T^{2} \)
29 \( 1 - 3.36T + 29T^{2} \)
31 \( 1 + 0.807T + 31T^{2} \)
37 \( 1 - 3.55iT - 37T^{2} \)
41 \( 1 + 9.22T + 41T^{2} \)
43 \( 1 - 4.15iT - 43T^{2} \)
47 \( 1 - 1.63iT - 47T^{2} \)
53 \( 1 + 7.56iT - 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 8.00iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 0.971iT - 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 3.33iT - 97T^{2} \)
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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

−10.25666575230035706655989401753, −9.702239250802259018229199911939, −8.466047819715390126166068516483, −8.201082883196613284444998392042, −6.79241701408279248198767550513, −5.73090990266327631451946830641, −5.17441795058289713027113817301, −3.95383052359677305625611436409, −3.13430414688615938606797761179, −2.53193085266868364548378760381, 0.45938641742697812528794078142, 1.55902735167032815216581757704, 2.62949475854978646467399879200, 3.87021046556502892049748024133, 5.18546471532398691487889737279, 6.46247210719223367315940649390, 6.89037749536510736720879531849, 7.47681536356353382345083092573, 8.101262260695432304558111947093, 8.770203087626323641894161262690

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