Properties

Label 2-1205-5.4-c1-0-2
Degree $2$
Conductor $1205$
Sign $0.852 + 0.523i$
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  − 2.68i·2-s − 2.58i·3-s − 5.21·4-s + (1.90 + 1.17i)5-s − 6.93·6-s + 1.24i·7-s + 8.64i·8-s − 3.66·9-s + (3.14 − 5.11i)10-s − 5.71·11-s + 13.4i·12-s + 3.28i·13-s + 3.35·14-s + (3.02 − 4.91i)15-s + 12.7·16-s + 5.29i·17-s + ⋯
L(s)  = 1  − 1.89i·2-s − 1.48i·3-s − 2.60·4-s + (0.852 + 0.523i)5-s − 2.83·6-s + 0.472i·7-s + 3.05i·8-s − 1.22·9-s + (0.994 − 1.61i)10-s − 1.72·11-s + 3.88i·12-s + 0.910i·13-s + 0.897·14-s + (0.779 − 1.26i)15-s + 3.19·16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.852 + 0.523i$
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: \((2,\ 1205,\ (\ :1/2),\ 0.852 + 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4008953523\)
\(L(\frac12)\) \(\approx\) \(0.4008953523\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.90 - 1.17i)T \)
241 \( 1 - T \)
good2 \( 1 + 2.68iT - 2T^{2} \)
3 \( 1 + 2.58iT - 3T^{2} \)
7 \( 1 - 1.24iT - 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
13 \( 1 - 3.28iT - 13T^{2} \)
17 \( 1 - 5.29iT - 17T^{2} \)
19 \( 1 + 6.14T + 19T^{2} \)
23 \( 1 + 2.22iT - 23T^{2} \)
29 \( 1 - 5.09T + 29T^{2} \)
31 \( 1 + 4.50T + 31T^{2} \)
37 \( 1 + 4.88iT - 37T^{2} \)
41 \( 1 + 4.77T + 41T^{2} \)
43 \( 1 + 2.72iT - 43T^{2} \)
47 \( 1 + 2.52iT - 47T^{2} \)
53 \( 1 - 7.67iT - 53T^{2} \)
59 \( 1 - 8.39T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 15.2iT - 67T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 - 5.27iT - 73T^{2} \)
79 \( 1 + 9.38T + 79T^{2} \)
83 \( 1 - 2.13iT - 83T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−10.12766801755649994478412766443, −8.782450605764301206079333356901, −8.480028112211517217097155405615, −7.29714233079373533968463683562, −6.18406133450388293704087479598, −5.42182997699387131887506190218, −4.17023308765552351015885012932, −2.70639982405774461162075739384, −2.27549586016676459570084184629, −1.55131188772437506304282825306, 0.16658388522123664529930408105, 3.02200475190051746686696847303, 4.35114428049398482021581645347, 5.03563804479865462734752852000, 5.35731372197044688312092539890, 6.29878849031496645048183869568, 7.39835701599845912458597938994, 8.233429386047127797214750219011, 8.841810914780805853387277711397, 9.715730498785871430968861894809

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