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Properties

Label 2-1170-65.18-c1-0-13
Degree $2$
Conductor $1170$
Sign $0.984 + 0.175i$
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 + (−2.20 + 0.376i)5-s − 1.65·7-s − i·8-s + (−0.376 − 2.20i)10-s + (−2.12 − 2.12i)11-s + (−0.310 + 3.59i)13-s − 1.65i·14-s + 16-s + (−0.877 − 0.877i)17-s + (3.06 + 3.06i)19-s + (2.20 − 0.376i)20-s + (2.12 − 2.12i)22-s + (2.10 − 2.10i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.985 + 0.168i)5-s − 0.626·7-s − 0.353i·8-s + (−0.118 − 0.697i)10-s + (−0.641 − 0.641i)11-s + (−0.0860 + 0.996i)13-s − 0.443i·14-s + 0.250·16-s + (−0.212 − 0.212i)17-s + (0.702 + 0.702i)19-s + (0.492 − 0.0841i)20-s + (0.453 − 0.453i)22-s + (0.439 − 0.439i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.984 + 0.175i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.984 + 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8453188946\)
\(L(\frac12)\) \(\approx\) \(0.8453188946\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 0.376i)T \)
13 \( 1 + (0.310 - 3.59i)T \)
good7 \( 1 + 1.65T + 7T^{2} \)
11 \( 1 + (2.12 + 2.12i)T + 11iT^{2} \)
17 \( 1 + (0.877 + 0.877i)T + 17iT^{2} \)
19 \( 1 + (-3.06 - 3.06i)T + 19iT^{2} \)
23 \( 1 + (-2.10 + 2.10i)T - 23iT^{2} \)
29 \( 1 + 4.94iT - 29T^{2} \)
31 \( 1 + (-6.61 + 6.61i)T - 31iT^{2} \)
37 \( 1 + 1.87T + 37T^{2} \)
41 \( 1 + (-6.17 + 6.17i)T - 41iT^{2} \)
43 \( 1 + (-5.76 + 5.76i)T - 43iT^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 + (6.92 + 6.92i)T + 53iT^{2} \)
59 \( 1 + (9.72 - 9.72i)T - 59iT^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 7.99iT - 67T^{2} \)
71 \( 1 + (-2.29 + 2.29i)T - 71iT^{2} \)
73 \( 1 - 3.90iT - 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (6.03 - 6.03i)T - 89iT^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−9.599558512190898967822570672477, −8.798926768181623128688290357099, −7.977488599098269820827213823573, −7.33057284845477937412040215691, −6.49341252249384436432244414108, −5.67886569887624747235524154636, −4.52043250177378022313949044737, −3.76148986767740723351691103088, −2.65556044887216515270703039698, −0.47567966054418854917076936095, 0.992795025831373927484220267464, 2.79392144767261404911870441054, 3.35107551846037867060181453360, 4.60088951887396629214086146281, 5.20618598631223276499714997419, 6.55845593325554512314178218212, 7.53771532609671385097425092258, 8.142337275733464588198252797046, 9.151530842892651776949795036269, 9.826399628262717233643350786807

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