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Properties

Label 2-4368-13.12-c1-0-20
Degree $2$
Conductor $4368$
Sign $0.358 - 0.933i$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
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  + 3-s − 0.660i·5-s − i·7-s + 9-s − 0.393i·11-s + (3.36 + 1.29i)13-s − 0.660i·15-s − 7.29·17-s + 6.32i·19-s − i·21-s − 5.56·23-s + 4.56·25-s + 27-s − 2.97·29-s + 8.46i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.295i·5-s − 0.377i·7-s + 0.333·9-s − 0.118i·11-s + (0.933 + 0.358i)13-s − 0.170i·15-s − 1.76·17-s + 1.44i·19-s − 0.218i·21-s − 1.16·23-s + 0.912·25-s + 0.192·27-s − 0.552·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4368} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 0.358 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867050123\)
\(L(\frac12)\) \(\approx\) \(1.867050123\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + iT \)
13 \( 1 + (-3.36 - 1.29i)T \)
good5 \( 1 + 0.660iT - 5T^{2} \)
11 \( 1 + 0.393iT - 11T^{2} \)
17 \( 1 + 7.29T + 17T^{2} \)
19 \( 1 - 6.32iT - 19T^{2} \)
23 \( 1 + 5.56T + 23T^{2} \)
29 \( 1 + 2.97T + 29T^{2} \)
31 \( 1 - 8.46iT - 31T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 + 0.998T + 43T^{2} \)
47 \( 1 + 6.04iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 1.07iT - 59T^{2} \)
61 \( 1 - 6.73T + 61T^{2} \)
67 \( 1 - 12.4iT - 67T^{2} \)
71 \( 1 - 8.39iT - 71T^{2} \)
73 \( 1 + 7.39iT - 73T^{2} \)
79 \( 1 - 5.41T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + 5.84iT - 89T^{2} \)
97 \( 1 + 6.22iT - 97T^{2} \)
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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

−8.577662329267653869874488888699, −8.010270693895279285132303066755, −6.93810203222888496831289805758, −6.51001102529893988013368339908, −5.58123636491320032053237813836, −4.56038951749262022482259061819, −3.97619855335907672867232823795, −3.19459260267378938256979060665, −2.04167464774647387637826924099, −1.23310275781118950151963878321, 0.48278791996653411404751578706, 2.12064714627358985013608228902, 2.51196056958608098890369845593, 3.70692337532661902528097414476, 4.27361066138417172050903441093, 5.27367947729427973496021604332, 6.15149341050618758459830452942, 6.79218000067181006360328714132, 7.52813985156030579468920127481, 8.307200964031568572318159535194

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