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Properties

Label 2-1205-5.4-c1-0-3
Degree $2$
Conductor $1205$
Sign $-0.438 - 0.898i$
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  − 1.88i·2-s + 2.55i·3-s − 1.54·4-s + (−0.979 − 2.00i)5-s + 4.81·6-s + 3.51i·7-s − 0.856i·8-s − 3.53·9-s + (−3.78 + 1.84i)10-s − 1.11·11-s − 3.95i·12-s − 0.230i·13-s + 6.62·14-s + (5.13 − 2.50i)15-s − 4.70·16-s + 4.35i·17-s + ⋯
L(s)  = 1  − 1.33i·2-s + 1.47i·3-s − 0.772·4-s + (−0.438 − 0.898i)5-s + 1.96·6-s + 1.32i·7-s − 0.302i·8-s − 1.17·9-s + (−1.19 + 0.583i)10-s − 0.336·11-s − 1.14i·12-s − 0.0639i·13-s + 1.77·14-s + (1.32 − 0.646i)15-s − 1.17·16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.438 - 0.898i$
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: \((2,\ 1205,\ (\ :1/2),\ -0.438 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4211026997\)
\(L(\frac12)\) \(\approx\) \(0.4211026997\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.979 + 2.00i)T \)
241 \( 1 - T \)
good2 \( 1 + 1.88iT - 2T^{2} \)
3 \( 1 - 2.55iT - 3T^{2} \)
7 \( 1 - 3.51iT - 7T^{2} \)
11 \( 1 + 1.11T + 11T^{2} \)
13 \( 1 + 0.230iT - 13T^{2} \)
17 \( 1 - 4.35iT - 17T^{2} \)
19 \( 1 + 3.14T + 19T^{2} \)
23 \( 1 + 6.36iT - 23T^{2} \)
29 \( 1 - 0.0527T + 29T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 - 6.27iT - 37T^{2} \)
41 \( 1 + 8.79T + 41T^{2} \)
43 \( 1 - 2.95iT - 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 6.03iT - 53T^{2} \)
59 \( 1 + 6.43T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 2.08iT - 67T^{2} \)
71 \( 1 + 5.17T + 71T^{2} \)
73 \( 1 - 1.25iT - 73T^{2} \)
79 \( 1 - 0.749T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 - 8.15T + 89T^{2} \)
97 \( 1 - 7.78iT - 97T^{2} \)
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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

−10.26345044615660308494114494975, −9.228532047651355924865271229088, −8.854520329948335807421068802673, −8.141547711658545747655599191280, −6.39272774880316667847885504608, −5.30772608071032320255801653697, −4.59840265191209155590627902913, −3.84857219783736938977615829543, −2.93869298814801828843596883362, −1.81281730430750949179465057821, 0.16834400652406978564631594016, 1.89049659484761856689386116246, 3.20046176106685770343809583384, 4.47855468681792676079161741978, 5.71026857405594508651787815919, 6.53656026734406092506318130326, 7.17583122511430348363185770712, 7.48849290887383356929939882445, 7.991197042837257420774401701599, 9.131791508730228512355618669507

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