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Properties

Label 2-1205-5.4-c1-0-100
Degree $2$
Conductor $1205$
Sign $-0.582 - 0.813i$
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  − 1.95i·2-s − 0.350i·3-s − 1.83·4-s + (−1.30 − 1.81i)5-s − 0.686·6-s + 1.14i·7-s − 0.318i·8-s + 2.87·9-s + (−3.56 + 2.54i)10-s + 0.395·11-s + 0.643i·12-s + 0.595i·13-s + 2.25·14-s + (−0.637 + 0.456i)15-s − 4.29·16-s − 7.79i·17-s + ⋯
L(s)  = 1  − 1.38i·2-s − 0.202i·3-s − 0.918·4-s + (−0.582 − 0.813i)5-s − 0.280·6-s + 0.434i·7-s − 0.112i·8-s + 0.959·9-s + (−1.12 + 0.806i)10-s + 0.119·11-s + 0.185i·12-s + 0.165i·13-s + 0.601·14-s + (−0.164 + 0.117i)15-s − 1.07·16-s − 1.88i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.582 - 0.813i$
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: \((2,\ 1205,\ (\ :1/2),\ -0.582 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.059550688\)
\(L(\frac12)\) \(\approx\) \(1.059550688\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.30 + 1.81i)T \)
241 \( 1 - T \)
good2 \( 1 + 1.95iT - 2T^{2} \)
3 \( 1 + 0.350iT - 3T^{2} \)
7 \( 1 - 1.14iT - 7T^{2} \)
11 \( 1 - 0.395T + 11T^{2} \)
13 \( 1 - 0.595iT - 13T^{2} \)
17 \( 1 + 7.79iT - 17T^{2} \)
19 \( 1 + 3.25T + 19T^{2} \)
23 \( 1 + 3.49iT - 23T^{2} \)
29 \( 1 + 6.23T + 29T^{2} \)
31 \( 1 + 9.01T + 31T^{2} \)
37 \( 1 - 0.667iT - 37T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 + 9.26iT - 43T^{2} \)
47 \( 1 - 13.6iT - 47T^{2} \)
53 \( 1 + 0.719iT - 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 2.84T + 61T^{2} \)
67 \( 1 - 1.17iT - 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 5.86iT - 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 0.175iT - 97T^{2} \)
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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

−9.220256888094635963136699998976, −8.889698071602687081204533749911, −7.53139603134103595127886825611, −6.97418491092466101039110215942, −5.53478029681014948890012302118, −4.49223421655785417751748505200, −3.91738026862681471035749908888, −2.66333704895349092367058662626, −1.66196606306809542014379752160, −0.44935309429584335899188621516, 1.94862072759299335260249945032, 3.74148316895231162086015488134, 4.17136628497980853248601622016, 5.49185192830774438817007158617, 6.28147072653607606279972994245, 7.09418688370479421190755523533, 7.56139523455228016659506335819, 8.318847915152071665803752031755, 9.242999013228017723920241495864, 10.34702530900879371856402474874

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