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Properties

Label 2-369-123.101-c1-0-6
Degree $2$
Conductor $369$
Sign $-0.330 - 0.943i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
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  + (0.814 + 1.59i)2-s + (−0.714 + 0.983i)4-s + (0.595 + 3.76i)5-s + (2.85 − 2.43i)7-s + (1.38 + 0.219i)8-s + (−5.52 + 4.01i)10-s + (−2.20 − 1.35i)11-s + (1.45 − 0.114i)13-s + (6.21 + 2.57i)14-s + (1.53 + 4.71i)16-s + (−3.62 + 0.870i)17-s + (−5.01 − 0.394i)19-s + (−4.12 − 2.10i)20-s + (0.364 − 4.63i)22-s + (1.36 − 4.18i)23-s + ⋯
L(s)  = 1  + (0.575 + 1.12i)2-s + (−0.357 + 0.491i)4-s + (0.266 + 1.68i)5-s + (1.07 − 0.920i)7-s + (0.490 + 0.0777i)8-s + (−1.74 + 1.26i)10-s + (−0.666 − 0.408i)11-s + (0.402 − 0.0316i)13-s + (1.66 + 0.687i)14-s + (0.382 + 1.17i)16-s + (−0.879 + 0.211i)17-s + (−1.14 − 0.0904i)19-s + (−0.922 − 0.470i)20-s + (0.0777 − 0.987i)22-s + (0.283 − 0.873i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ -0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20537 + 1.69904i\)
\(L(\frac12)\) \(\approx\) \(1.20537 + 1.69904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (6.04 - 2.11i)T \)
good2 \( 1 + (-0.814 - 1.59i)T + (-1.17 + 1.61i)T^{2} \)
5 \( 1 + (-0.595 - 3.76i)T + (-4.75 + 1.54i)T^{2} \)
7 \( 1 + (-2.85 + 2.43i)T + (1.09 - 6.91i)T^{2} \)
11 \( 1 + (2.20 + 1.35i)T + (4.99 + 9.80i)T^{2} \)
13 \( 1 + (-1.45 + 0.114i)T + (12.8 - 2.03i)T^{2} \)
17 \( 1 + (3.62 - 0.870i)T + (15.1 - 7.71i)T^{2} \)
19 \( 1 + (5.01 + 0.394i)T + (18.7 + 2.97i)T^{2} \)
23 \( 1 + (-1.36 + 4.18i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.50 - 0.601i)T + (25.8 + 13.1i)T^{2} \)
31 \( 1 + (4.04 + 5.56i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.62 - 6.26i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (0.560 - 0.285i)T + (25.2 - 34.7i)T^{2} \)
47 \( 1 + (-7.81 + 9.15i)T + (-7.35 - 46.4i)T^{2} \)
53 \( 1 + (-1.07 + 4.46i)T + (-47.2 - 24.0i)T^{2} \)
59 \( 1 + (11.4 + 3.73i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.85 + 5.60i)T + (-35.8 - 49.3i)T^{2} \)
67 \( 1 + (-10.1 + 6.19i)T + (30.4 - 59.6i)T^{2} \)
71 \( 1 + (0.295 - 0.482i)T + (-32.2 - 63.2i)T^{2} \)
73 \( 1 + (1.50 + 1.50i)T + 73iT^{2} \)
79 \( 1 + (4.52 - 1.87i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 - 7.97iT - 83T^{2} \)
89 \( 1 + (-9.86 - 11.5i)T + (-13.9 + 87.9i)T^{2} \)
97 \( 1 + (1.77 + 2.89i)T + (-44.0 + 86.4i)T^{2} \)
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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

−11.20969680650351353009964808226, −10.87269453077074521991257589711, −10.21979656069921958917688052433, −8.379611240207147759094628930428, −7.67857365616874345190030232166, −6.72126730013544463428720327306, −6.24037782460038926720519236009, −4.91012634751489866047632642222, −3.87968017580771377578091794503, −2.24653946974150099593787131359, 1.48300153574019981342320352857, 2.39577601183602663414686427008, 4.25759892783539385074888200786, 4.88469189389889968323145546848, 5.71240953352104340556611269314, 7.62932276048768646249626036625, 8.627021874699506906516748137615, 9.212794067241586740312314586131, 10.51684454856848232724677925614, 11.34270657948537266318660305294

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