Properties

Label 2-4368-13.12-c1-0-38
Degree $2$
Conductor $4368$
Sign $-0.0276 - 0.999i$
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 + 3.22i·5-s i·7-s + 9-s + 5.74i·11-s + (3.60 − 0.0996i)13-s + 3.22i·15-s + 4.91·17-s + 1.28i·19-s i·21-s + 4.39·23-s − 5.39·25-s + 27-s + 4.19·29-s + 6.68i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.44i·5-s − 0.377i·7-s + 0.333·9-s + 1.73i·11-s + (0.999 − 0.0276i)13-s + 0.832i·15-s + 1.19·17-s + 0.293i·19-s − 0.218i·21-s + 0.916·23-s − 1.07·25-s + 0.192·27-s + 0.779·29-s + 1.20i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0276 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0276 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0276 - 0.999i$
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: \((2,\ 4368,\ (\ :1/2),\ -0.0276 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.723667876\)
\(L(\frac12)\) \(\approx\) \(2.723667876\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0996i)T \)
good5 \( 1 - 3.22iT - 5T^{2} \)
11 \( 1 - 5.74iT - 11T^{2} \)
17 \( 1 - 4.91T + 17T^{2} \)
19 \( 1 - 1.28iT - 19T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 - 4.19T + 29T^{2} \)
31 \( 1 - 6.68iT - 31T^{2} \)
37 \( 1 + 5.95iT - 37T^{2} \)
41 \( 1 + 9.82iT - 41T^{2} \)
43 \( 1 + 3.72T + 43T^{2} \)
47 \( 1 + 0.506iT - 47T^{2} \)
53 \( 1 + 5.63T + 53T^{2} \)
59 \( 1 - 2.70iT - 59T^{2} \)
61 \( 1 - 7.20T + 61T^{2} \)
67 \( 1 + 5.31iT - 67T^{2} \)
71 \( 1 - 2.25iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 7.80T + 79T^{2} \)
83 \( 1 + 9.42iT - 83T^{2} \)
89 \( 1 + 9.79iT - 89T^{2} \)
97 \( 1 - 9.36iT - 97T^{2} \)
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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

−8.483795617582584095624971820888, −7.63177824046120232254347596847, −7.08583209812641135869824386254, −6.71605582734359602834199206507, −5.68279266480115045762169898132, −4.70979013457645245846688387125, −3.74501313783881484082598587207, −3.22372062154012519686055538017, −2.30971378703081680564548977606, −1.34995075441075808394433891654, 0.804030790202150393425481040144, 1.39047483801555103182149490615, 2.88482031003029753735817449909, 3.45917382159314916272556606080, 4.44646443893296401876985423889, 5.23771542741522476808046853774, 5.88347762271369652899228522450, 6.59048333938249824836689491109, 7.935678682190949629920838734923, 8.261124237395135017932957809500

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