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Properties

Label 2-1205-5.4-c1-0-21
Degree $2$
Conductor $1205$
Sign $-0.673 - 0.739i$
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  + 1.09i·2-s − 0.223i·3-s + 0.800·4-s + (−1.50 − 1.65i)5-s + 0.245·6-s + 2.26i·7-s + 3.06i·8-s + 2.94·9-s + (1.81 − 1.64i)10-s − 3.16·11-s − 0.179i·12-s + 0.0649i·13-s − 2.47·14-s + (−0.370 + 0.337i)15-s − 1.75·16-s + 4.44i·17-s + ⋯
L(s)  = 1  + 0.774i·2-s − 0.129i·3-s + 0.400·4-s + (−0.673 − 0.739i)5-s + 0.100·6-s + 0.855i·7-s + 1.08i·8-s + 0.983·9-s + (0.572 − 0.521i)10-s − 0.954·11-s − 0.0517i·12-s + 0.0180i·13-s − 0.662·14-s + (−0.0956 + 0.0870i)15-s − 0.439·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.673 - 0.739i$
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: \((2,\ 1205,\ (\ :1/2),\ -0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.371589833\)
\(L(\frac12)\) \(\approx\) \(1.371589833\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.50 + 1.65i)T \)
241 \( 1 - T \)
good2 \( 1 - 1.09iT - 2T^{2} \)
3 \( 1 + 0.223iT - 3T^{2} \)
7 \( 1 - 2.26iT - 7T^{2} \)
11 \( 1 + 3.16T + 11T^{2} \)
13 \( 1 - 0.0649iT - 13T^{2} \)
17 \( 1 - 4.44iT - 17T^{2} \)
19 \( 1 + 7.82T + 19T^{2} \)
23 \( 1 - 2.53iT - 23T^{2} \)
29 \( 1 - 6.53T + 29T^{2} \)
31 \( 1 - 8.27T + 31T^{2} \)
37 \( 1 + 2.73iT - 37T^{2} \)
41 \( 1 + 7.52T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 - 8.53iT - 47T^{2} \)
53 \( 1 + 0.0654iT - 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 + 8.56T + 71T^{2} \)
73 \( 1 - 16.6iT - 73T^{2} \)
79 \( 1 - 0.725T + 79T^{2} \)
83 \( 1 - 6.57iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 9.64iT - 97T^{2} \)
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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

−10.06372783163003139942389286349, −8.834169580463599740207355921320, −8.176297575533939458042888546823, −7.79176304488756034490471873914, −6.63982690036667186071717812994, −6.04958146465064132171879866962, −4.98353175372926675879787375470, −4.29168950510076588205198131791, −2.76351519222858881421523019751, −1.63080702202679432132407664843, 0.56625514274517039467906464554, 2.18894655331157736920072613715, 3.06994235229316087779212479834, 4.07365268016518032842154435953, 4.73521903179736938151960486526, 6.49785960222748092334499379478, 6.92107312113077677968770888756, 7.66744793654969578767536150430, 8.591457409906122130697037882477, 10.06394124000071662204643981604

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