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Properties

Label 2-1160-145.12-c1-0-15
Degree $2$
Conductor $1160$
Sign $0.977 - 0.209i$
Analytic cond. $9.26264$
Root an. cond. $3.04345$
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  − 1.24·3-s + (1.73 − 1.40i)5-s + (−1.01 − 1.01i)7-s − 1.44·9-s + (4.38 + 4.38i)11-s + (0.898 + 0.898i)13-s + (−2.16 + 1.75i)15-s + 1.76i·17-s + (−3.91 + 3.91i)19-s + (1.26 + 1.26i)21-s + (0.505 − 0.505i)23-s + (1.04 − 4.89i)25-s + 5.54·27-s + (5.04 − 1.88i)29-s + (4.04 + 4.04i)31-s + ⋯
L(s)  = 1  − 0.719·3-s + (0.777 − 0.629i)5-s + (−0.384 − 0.384i)7-s − 0.483·9-s + (1.32 + 1.32i)11-s + (0.249 + 0.249i)13-s + (−0.558 + 0.452i)15-s + 0.428i·17-s + (−0.898 + 0.898i)19-s + (0.276 + 0.276i)21-s + (0.105 − 0.105i)23-s + (0.208 − 0.978i)25-s + 1.06·27-s + (0.936 − 0.349i)29-s + (0.726 + 0.726i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(9.26264\)
Root analytic conductor: \(3.04345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1160} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1160,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.392075214\)
\(L(\frac12)\) \(\approx\) \(1.392075214\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.73 + 1.40i)T \)
29 \( 1 + (-5.04 + 1.88i)T \)
good3 \( 1 + 1.24T + 3T^{2} \)
7 \( 1 + (1.01 + 1.01i)T + 7iT^{2} \)
11 \( 1 + (-4.38 - 4.38i)T + 11iT^{2} \)
13 \( 1 + (-0.898 - 0.898i)T + 13iT^{2} \)
17 \( 1 - 1.76iT - 17T^{2} \)
19 \( 1 + (3.91 - 3.91i)T - 19iT^{2} \)
23 \( 1 + (-0.505 + 0.505i)T - 23iT^{2} \)
31 \( 1 + (-4.04 - 4.04i)T + 31iT^{2} \)
37 \( 1 - 1.45T + 37T^{2} \)
41 \( 1 + (-6.26 + 6.26i)T - 41iT^{2} \)
43 \( 1 + 7.71T + 43T^{2} \)
47 \( 1 - 6.29T + 47T^{2} \)
53 \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \)
59 \( 1 - 14.6iT - 59T^{2} \)
61 \( 1 + (-5.27 - 5.27i)T + 61iT^{2} \)
67 \( 1 + (-4.62 + 4.62i)T - 67iT^{2} \)
71 \( 1 - 3.20iT - 71T^{2} \)
73 \( 1 - 4.68iT - 73T^{2} \)
79 \( 1 + (8.74 - 8.74i)T - 79iT^{2} \)
83 \( 1 + (-9.25 + 9.25i)T - 83iT^{2} \)
89 \( 1 + (3.02 - 3.02i)T - 89iT^{2} \)
97 \( 1 + 6.99T + 97T^{2} \)
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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

−10.08945097344610677156213950631, −8.961310903160764620833587474057, −8.449216069974114404967169720186, −6.99903546971868079019456513891, −6.40787549139567097805602918217, −5.71587919668703171696378119693, −4.63985841755737195037960469231, −3.92654643307296935965775397038, −2.24672962564533748182676657821, −1.08755078181893692449392883362, 0.838355394918631697821555700779, 2.55345643466423836170762455255, 3.36884233362810596298514295001, 4.75120534071571687467021219624, 5.89137593484138453691220445799, 6.22980593443082743698699337234, 6.85892601507063259740998422459, 8.334274311817733508607302193697, 9.007297598051919512458776041467, 9.737119800954645052126653763282

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