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Properties

Label 2-1205-5.4-c1-0-26
Degree $2$
Conductor $1205$
Sign $-0.999 + 0.0172i$
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  + 0.930i·2-s + 1.32i·3-s + 1.13·4-s + (−2.23 + 0.0386i)5-s − 1.23·6-s + 4.33i·7-s + 2.91i·8-s + 1.24·9-s + (−0.0359 − 2.07i)10-s + 0.882·11-s + 1.50i·12-s + 2.68i·13-s − 4.03·14-s + (−0.0512 − 2.96i)15-s − 0.442·16-s − 0.304i·17-s + ⋯
L(s)  = 1  + 0.657i·2-s + 0.765i·3-s + 0.567·4-s + (−0.999 + 0.0172i)5-s − 0.503·6-s + 1.63i·7-s + 1.03i·8-s + 0.413·9-s + (−0.0113 − 0.657i)10-s + 0.266·11-s + 0.434i·12-s + 0.743i·13-s − 1.07·14-s + (−0.0132 − 0.765i)15-s − 0.110·16-s − 0.0738i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.999 + 0.0172i$
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: \((2,\ 1205,\ (\ :1/2),\ -0.999 + 0.0172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.654978065\)
\(L(\frac12)\) \(\approx\) \(1.654978065\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 - 0.0386i)T \)
241 \( 1 - T \)
good2 \( 1 - 0.930iT - 2T^{2} \)
3 \( 1 - 1.32iT - 3T^{2} \)
7 \( 1 - 4.33iT - 7T^{2} \)
11 \( 1 - 0.882T + 11T^{2} \)
13 \( 1 - 2.68iT - 13T^{2} \)
17 \( 1 + 0.304iT - 17T^{2} \)
19 \( 1 - 4.13T + 19T^{2} \)
23 \( 1 + 2.19iT - 23T^{2} \)
29 \( 1 + 8.97T + 29T^{2} \)
31 \( 1 - 5.91T + 31T^{2} \)
37 \( 1 + 10.9iT - 37T^{2} \)
41 \( 1 + 6.07T + 41T^{2} \)
43 \( 1 + 2.71iT - 43T^{2} \)
47 \( 1 + 9.50iT - 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 - 7.93T + 59T^{2} \)
61 \( 1 + 1.60T + 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 + 4.11iT - 73T^{2} \)
79 \( 1 + 9.90T + 79T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 - 9.48T + 89T^{2} \)
97 \( 1 - 1.41iT - 97T^{2} \)
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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

−10.08836383321975666255324249679, −9.008801239053248769227701585347, −8.697752896074209070078655977286, −7.53126495488492367313269806355, −6.99394416947257734895585878012, −5.87961095157721219724634087783, −5.20109036090849218275847261219, −4.20004819688069287535381391332, −3.14218528347521505853626561741, −1.98419345276183775287096380090, 0.74599834087566876078893960918, 1.51180179760057382904742454944, 3.15512618839718198885785375061, 3.76318724814692510564327322182, 4.74964373355275329373230311027, 6.34251555863254811282802505657, 7.08621196834159246441431470625, 7.55729223597961627617243998468, 8.127554096448996027505218717450, 9.737329092926165182576974020952

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