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Properties

Label 2-1368-1368.1067-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.342 + 0.939i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s − 3-s + (−0.939 − 0.342i)4-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)8-s + 9-s + 1.28i·11-s + (0.939 + 0.342i)12-s + (0.766 + 0.642i)16-s + (1.11 − 1.32i)17-s + (0.173 − 0.984i)18-s + (0.939 + 0.342i)19-s + (1.26 + 0.223i)22-s + (0.5 − 0.866i)24-s + (−0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s − 3-s + (−0.939 − 0.342i)4-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)8-s + 9-s + 1.28i·11-s + (0.939 + 0.342i)12-s + (0.766 + 0.642i)16-s + (1.11 − 1.32i)17-s + (0.173 − 0.984i)18-s + (0.939 + 0.342i)19-s + (1.26 + 0.223i)22-s + (0.5 − 0.866i)24-s + (−0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8016015706\)
\(L(\frac12)\) \(\approx\) \(0.8016015706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good5 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 - 1.28iT - T^{2} \)
13 \( 1 + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
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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

−9.827988024321944433758055840042, −9.354603206407527304639542606683, −7.933354768427975036712586882167, −7.21223876325451445431681001191, −6.09771884934781324282359397284, −5.16241690347600198825276577203, −4.65983177685473246492024517130, −3.58445489712590700540896690553, −2.31532449828612221869500516391, −1.03046867023738215208466821681, 1.06824133410458858643937277153, 3.33295228093235711734059214919, 4.12343823278655515390973869361, 5.41349031288206695145463319191, 5.66879016393376835621794877960, 6.50926498453366966814249452942, 7.44572072782062578012628653461, 8.079681021261766498520577683591, 9.080517828765379451603313051059, 9.839693139671879205298101822632

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