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Properties

Label 2-1155-5.4-c1-0-46
Degree $2$
Conductor $1155$
Sign $-0.437 + 0.899i$
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  − 0.437i·2-s − i·3-s + 1.80·4-s + (0.978 − 2.01i)5-s − 0.437·6-s + i·7-s − 1.66i·8-s − 9-s + (−0.879 − 0.427i)10-s − 11-s − 1.80i·12-s + 0.0257i·13-s + 0.437·14-s + (−2.01 − 0.978i)15-s + 2.88·16-s − 1.73i·17-s + ⋯
L(s)  = 1  − 0.309i·2-s − 0.577i·3-s + 0.904·4-s + (0.437 − 0.899i)5-s − 0.178·6-s + 0.377i·7-s − 0.588i·8-s − 0.333·9-s + (−0.278 − 0.135i)10-s − 0.301·11-s − 0.522i·12-s + 0.00715i·13-s + 0.116·14-s + (−0.519 − 0.252i)15-s + 0.722·16-s − 0.421i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053895275\)
\(L(\frac12)\) \(\approx\) \(2.053895275\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.978 + 2.01i)T \)
7 \( 1 - iT \)
11 \( 1 + T \)
good2 \( 1 + 0.437iT - 2T^{2} \)
13 \( 1 - 0.0257iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 2.62T + 19T^{2} \)
23 \( 1 + 8.28iT - 23T^{2} \)
29 \( 1 - 4.93T + 29T^{2} \)
31 \( 1 - 1.90T + 31T^{2} \)
37 \( 1 + 3.18iT - 37T^{2} \)
41 \( 1 + 2.07T + 41T^{2} \)
43 \( 1 - 3.85iT - 43T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 - 7.00T + 59T^{2} \)
61 \( 1 + 1.71T + 61T^{2} \)
67 \( 1 - 7.39iT - 67T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 - 8.85T + 79T^{2} \)
83 \( 1 - 9.51iT - 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + 0.755iT - 97T^{2} \)
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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

−9.611764697353643031732929862881, −8.549058696529037217926297139791, −8.060792650750298604609245563337, −6.86340261486798549975347343997, −6.28515302692650870651786313111, −5.37394538965018005232066798918, −4.33959632112139530848426611025, −2.79791908385815243840639316962, −2.11268173890255394787398042860, −0.868345623384620247349322226764, 1.81283447828126712503772931727, 2.90504839748132821783928442001, 3.75300448002468006224029430874, 5.12442171663904031492351959397, 5.97638966630069355239975755214, 6.68907160426606160649105329957, 7.47202941395774806657005688890, 8.263087342664547786680964454455, 9.393531334151384813885477744027, 10.38760826583372933887476653022

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