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Properties

Label 2-1155-77.76-c1-0-32
Degree $2$
Conductor $1155$
Sign $0.756 + 0.653i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  − 1.54i·2-s + i·3-s − 0.393·4-s + i·5-s + 1.54·6-s + (2.51 − 0.814i)7-s − 2.48i·8-s − 9-s + 1.54·10-s + (1.71 + 2.83i)11-s − 0.393i·12-s + 2.83·13-s + (−1.26 − 3.89i)14-s − 15-s − 4.63·16-s + 1.11·17-s + ⋯
L(s)  = 1  − 1.09i·2-s + 0.577i·3-s − 0.196·4-s + 0.447i·5-s + 0.631·6-s + (0.951 − 0.308i)7-s − 0.878i·8-s − 0.333·9-s + 0.489·10-s + (0.518 + 0.855i)11-s − 0.113i·12-s + 0.785·13-s + (−0.336 − 1.04i)14-s − 0.258·15-s − 1.15·16-s + 0.269·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.756 + 0.653i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.756 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.111164699\)
\(L(\frac12)\) \(\approx\) \(2.111164699\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 - iT \)
7 \( 1 + (-2.51 + 0.814i)T \)
11 \( 1 + (-1.71 - 2.83i)T \)
good2 \( 1 + 1.54iT - 2T^{2} \)
13 \( 1 - 2.83T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + 4.41T + 23T^{2} \)
29 \( 1 - 2.50iT - 29T^{2} \)
31 \( 1 - 4.63iT - 31T^{2} \)
37 \( 1 - 3.30T + 37T^{2} \)
41 \( 1 - 4.46T + 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 - 3.19iT - 47T^{2} \)
53 \( 1 + 3.06T + 53T^{2} \)
59 \( 1 + 4.55iT - 59T^{2} \)
61 \( 1 - 2.53T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 3.76T + 71T^{2} \)
73 \( 1 - 4.86T + 73T^{2} \)
79 \( 1 + 4.77iT - 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 - 0.588iT - 89T^{2} \)
97 \( 1 - 7.73iT - 97T^{2} \)
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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

−9.903686776629468658798908594727, −9.257134073497488367945304081628, −8.171477762284410864040558296822, −7.25656332046411284781884267113, −6.37289415304614849859480762505, −5.15544757338356881057978646966, −4.12744352419009109408379647027, −3.52206357655019473920296189905, −2.28554483733400494354672459884, −1.26615352488553429889154016573, 1.17486902581707544379442531831, 2.40302380448107986376230500420, 3.93521800142714244141348470775, 5.12690479524490834533124144581, 5.89614334985202569774908229930, 6.38934679435030758389849158518, 7.59672822986723664808740288840, 8.074564230537139479532109340806, 8.645440266874491958566299750788, 9.536688448435642775893316664282

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