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Properties

Label 2-369-41.33-c1-0-1
Degree $2$
Conductor $369$
Sign $-0.797 + 0.603i$
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  + (−1.42 + 1.96i)2-s + (−1.20 − 3.69i)4-s + (−0.110 + 0.0358i)5-s + (−0.422 + 2.66i)7-s + (4.36 + 1.41i)8-s + (0.0870 − 0.267i)10-s + (1.53 + 3.01i)11-s + (−1.03 + 0.164i)13-s + (−4.63 − 4.63i)14-s + (−2.71 + 1.97i)16-s + (−3.25 + 1.66i)17-s + (−2.25 − 0.356i)19-s + (0.265 + 0.365i)20-s + (−8.11 − 1.28i)22-s + (−6.06 − 4.40i)23-s + ⋯
L(s)  = 1  + (−1.00 + 1.38i)2-s + (−0.601 − 1.84i)4-s + (−0.0493 + 0.0160i)5-s + (−0.159 + 1.00i)7-s + (1.54 + 0.501i)8-s + (0.0275 − 0.0847i)10-s + (0.463 + 0.909i)11-s + (−0.288 + 0.0456i)13-s + (−1.23 − 1.23i)14-s + (−0.678 + 0.492i)16-s + (−0.790 + 0.402i)17-s + (−0.516 − 0.0818i)19-s + (0.0593 + 0.0817i)20-s + (−1.72 − 0.273i)22-s + (−1.26 − 0.918i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $-0.797 + 0.603i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ -0.797 + 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131664 - 0.392153i\)
\(L(\frac12)\) \(\approx\) \(0.131664 - 0.392153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 6.26i)T \)
good2 \( 1 + (1.42 - 1.96i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.110 - 0.0358i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.422 - 2.66i)T + (-6.65 - 2.16i)T^{2} \)
11 \( 1 + (-1.53 - 3.01i)T + (-6.46 + 8.89i)T^{2} \)
13 \( 1 + (1.03 - 0.164i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (3.25 - 1.66i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (2.25 + 0.356i)T + (18.0 + 5.87i)T^{2} \)
23 \( 1 + (6.06 + 4.40i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.31 + 0.669i)T + (17.0 + 23.4i)T^{2} \)
31 \( 1 + (0.964 - 2.96i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.348 + 1.07i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.583 + 0.803i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.73 - 10.9i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.482 + 0.246i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.57 - 1.14i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.95 + 8.20i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (6.02 - 11.8i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-4.28 - 8.40i)T + (-41.7 + 57.4i)T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 + (-7.58 + 7.58i)T - 79iT^{2} \)
83 \( 1 - 0.635T + 83T^{2} \)
89 \( 1 + (0.753 - 4.75i)T + (-84.6 - 27.5i)T^{2} \)
97 \( 1 + (1.25 - 2.46i)T + (-57.0 - 78.4i)T^{2} \)
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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

−11.98807643368217015290730313118, −10.67016299326249605788280602204, −9.684444902786227908492965080859, −9.041294587568608168858707825986, −8.266103603610536350696746306699, −7.27928665780212536119186359241, −6.39608228795302709296615270679, −5.62567403131559447061316250281, −4.32788111460314177573274924759, −2.09633094016977009427592509839, 0.37607269810315525729101413927, 1.95851139007069505449456071548, 3.41554236069908792974351480821, 4.25606848231011185415233773709, 6.13936005633857538949707195909, 7.47638296306951221178038066423, 8.312226880138771125798237382109, 9.283451189306712035597229853237, 10.02871487226981696664154080598, 10.80951475273329114718372979998

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