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Properties

Label 2-1205-5.4-c1-0-106
Degree $2$
Conductor $1205$
Sign $-0.967 - 0.252i$
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  − 0.260i·2-s − 1.85i·3-s + 1.93·4-s + (−2.16 − 0.563i)5-s − 0.484·6-s + 1.27i·7-s − 1.02i·8-s − 0.456·9-s + (−0.146 + 0.563i)10-s − 5.63·11-s − 3.59i·12-s − 6.25i·13-s + 0.332·14-s + (−1.04 + 4.02i)15-s + 3.59·16-s − 0.195i·17-s + ⋯
L(s)  = 1  − 0.184i·2-s − 1.07i·3-s + 0.966·4-s + (−0.967 − 0.252i)5-s − 0.197·6-s + 0.481i·7-s − 0.362i·8-s − 0.152·9-s + (−0.0464 + 0.178i)10-s − 1.69·11-s − 1.03i·12-s − 1.73i·13-s + 0.0887·14-s + (−0.270 + 1.03i)15-s + 0.899·16-s − 0.0473i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.967 - 0.252i$
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: \((2,\ 1205,\ (\ :1/2),\ -0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8380461508\)
\(L(\frac12)\) \(\approx\) \(0.8380461508\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.16 + 0.563i)T \)
241 \( 1 - T \)
good2 \( 1 + 0.260iT - 2T^{2} \)
3 \( 1 + 1.85iT - 3T^{2} \)
7 \( 1 - 1.27iT - 7T^{2} \)
11 \( 1 + 5.63T + 11T^{2} \)
13 \( 1 + 6.25iT - 13T^{2} \)
17 \( 1 + 0.195iT - 17T^{2} \)
19 \( 1 + 3.74T + 19T^{2} \)
23 \( 1 - 8.06iT - 23T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 7.12iT - 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 + 4.60iT - 43T^{2} \)
47 \( 1 + 8.43iT - 47T^{2} \)
53 \( 1 - 8.07iT - 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 - 6.67iT - 67T^{2} \)
71 \( 1 + 4.71T + 71T^{2} \)
73 \( 1 + 9.61iT - 73T^{2} \)
79 \( 1 - 5.87T + 79T^{2} \)
83 \( 1 - 3.59iT - 83T^{2} \)
89 \( 1 + 0.879T + 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−9.179702722817101691943680154136, −8.022416701230765278892056271749, −7.53299163523851735271612193229, −7.30959661692677300994452219891, −5.78271304109220980048576785932, −5.43596434364250387450166669045, −3.72533752712783180093306598405, −2.77403145856534540489261241852, −1.83875927304202263625597711408, −0.32075984336173264914152028824, 2.09748049645900360376851490388, 3.21702995857055321099278945087, 4.23825442034209235794603243969, 4.79436860996297150672409705362, 6.09049213017882366956145510368, 7.04224065679199286411282810908, 7.58896358720740559665940668452, 8.482037856658249630432868572452, 9.458772313228104415152421917933, 10.50446415227489295437107515904

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