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Properties

Label 2-1170-65.47-c1-0-2
Degree $2$
Conductor $1170$
Sign $0.112 - 0.993i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  − i·2-s − 4-s + (0.372 − 2.20i)5-s + 1.64·7-s + i·8-s + (−2.20 − 0.372i)10-s + (−4.51 + 4.51i)11-s + (−0.732 + 3.53i)13-s − 1.64i·14-s + 16-s + (−5.07 + 5.07i)17-s + (−1.47 + 1.47i)19-s + (−0.372 + 2.20i)20-s + (4.51 + 4.51i)22-s + (−5.93 − 5.93i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.166 − 0.986i)5-s + 0.620·7-s + 0.353i·8-s + (−0.697 − 0.117i)10-s + (−1.36 + 1.36i)11-s + (−0.203 + 0.979i)13-s − 0.438i·14-s + 0.250·16-s + (−1.22 + 1.22i)17-s + (−0.338 + 0.338i)19-s + (−0.0832 + 0.493i)20-s + (0.963 + 0.963i)22-s + (−1.23 − 1.23i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4951021909\)
\(L(\frac12)\) \(\approx\) \(0.4951021909\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-0.372 + 2.20i)T \)
13 \( 1 + (0.732 - 3.53i)T \)
good7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 + (4.51 - 4.51i)T - 11iT^{2} \)
17 \( 1 + (5.07 - 5.07i)T - 17iT^{2} \)
19 \( 1 + (1.47 - 1.47i)T - 19iT^{2} \)
23 \( 1 + (5.93 + 5.93i)T + 23iT^{2} \)
29 \( 1 - 4.59iT - 29T^{2} \)
31 \( 1 + (-1.29 - 1.29i)T + 31iT^{2} \)
37 \( 1 - 2.57T + 37T^{2} \)
41 \( 1 + (6.40 + 6.40i)T + 41iT^{2} \)
43 \( 1 + (-1.01 - 1.01i)T + 43iT^{2} \)
47 \( 1 - 7.69T + 47T^{2} \)
53 \( 1 + (-2.67 + 2.67i)T - 53iT^{2} \)
59 \( 1 + (-1.77 - 1.77i)T + 59iT^{2} \)
61 \( 1 + 6.19T + 61T^{2} \)
67 \( 1 - 8.73iT - 67T^{2} \)
71 \( 1 + (1.40 + 1.40i)T + 71iT^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 9.18T + 83T^{2} \)
89 \( 1 + (1.46 + 1.46i)T + 89iT^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−10.23441343525214378444137384098, −9.114791685931928705862490779502, −8.488484311003167720341914357403, −7.79560402688645039770702402663, −6.62016663859238383007085163100, −5.43862418717581217516423584975, −4.53154257078017983037164423757, −4.20017184759967014305421015208, −2.24591775955587912507491757549, −1.79739765409175471259981412392, 0.19708065077722579757634680555, 2.40314454917811175667800291467, 3.25188517334847006643293591940, 4.59177813249450237104386938095, 5.57139998680858106405338252395, 6.11077574379331773066727890468, 7.24669748590740376192738095617, 7.86397141156953993749172725976, 8.474508530364979352139014676075, 9.621816236870240074722421294982

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