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Properties

Label 2-364-13.9-c1-0-1
Degree $2$
Conductor $364$
Sign $0.859 - 0.511i$
Analytic cond. $2.90655$
Root an. cond. $1.70486$
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  − 5-s + (0.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + (1 − 1.73i)11-s + (3.5 + 0.866i)13-s + (1.5 + 2.59i)17-s + (3 + 5.19i)19-s + (2 − 3.46i)23-s − 4·25-s + (3.5 − 6.06i)29-s + 4·31-s + (−0.5 − 0.866i)35-s + (−4.5 + 7.79i)37-s + (−4.5 + 7.79i)41-s + (−5 − 8.66i)43-s + ⋯
L(s)  = 1  − 0.447·5-s + (0.188 + 0.327i)7-s + (0.5 + 0.866i)9-s + (0.301 − 0.522i)11-s + (0.970 + 0.240i)13-s + (0.363 + 0.630i)17-s + (0.688 + 1.19i)19-s + (0.417 − 0.722i)23-s − 0.800·25-s + (0.649 − 1.12i)29-s + 0.718·31-s + (−0.0845 − 0.146i)35-s + (−0.739 + 1.28i)37-s + (−0.702 + 1.21i)41-s + (−0.762 − 1.32i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364\)    =    \(2^{2} \cdot 7 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(2.90655\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{364} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 364,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32073 + 0.362980i\)
\(L(\frac12)\) \(\approx\) \(1.32073 + 0.362980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5 + 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.60559186698398431116290080569, −10.57187929245578525278766899395, −9.822632282135129075629780317203, −8.319069613780472477798336206496, −8.127301024352791402083729070614, −6.68977739888406436917178181352, −5.69796008771547316423098302330, −4.46066407150155076441769736227, −3.36039004425604906387190846365, −1.60441124135414608045690655263, 1.14656171956604921705932898900, 3.21778095408038641361974355011, 4.20912539731422956209975365107, 5.41825273629224614893122090096, 6.81533664913621079047290992508, 7.38177118936365565035624442487, 8.656021550420366890114093580093, 9.471408403704956456003547074650, 10.43117212952993506618170516742, 11.49862052475461955828631029153

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