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Properties

Label 2-370-185.23-c1-0-10
Degree $2$
Conductor $370$
Sign $-0.419 + 0.907i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (−0.866 − 0.5i)2-s + (−1.69 − 0.453i)3-s + (0.499 + 0.866i)4-s + (−1.12 − 1.93i)5-s + (1.24 + 1.24i)6-s + (4.67 + 1.25i)7-s − 0.999i·8-s + (0.0652 + 0.0376i)9-s + (0.0112 + 2.23i)10-s + 1.80i·11-s + (−0.453 − 1.69i)12-s + (2.26 − 1.30i)13-s + (−3.42 − 3.42i)14-s + (1.03 + 3.78i)15-s + (−0.5 + 0.866i)16-s + (1.73 − 3.01i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.977 − 0.262i)3-s + (0.249 + 0.433i)4-s + (−0.504 − 0.863i)5-s + (0.506 + 0.506i)6-s + (1.76 + 0.473i)7-s − 0.353i·8-s + (0.0217 + 0.0125i)9-s + (0.00354 + 0.707i)10-s + 0.544i·11-s + (−0.131 − 0.488i)12-s + (0.628 − 0.362i)13-s + (−0.914 − 0.914i)14-s + (0.266 + 0.976i)15-s + (−0.125 + 0.216i)16-s + (0.421 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358536 - 0.560795i\)
\(L(\frac12)\) \(\approx\) \(0.358536 - 0.560795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (1.12 + 1.93i)T \)
37 \( 1 + (5.65 + 2.23i)T \)
good3 \( 1 + (1.69 + 0.453i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-4.67 - 1.25i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 - 1.80iT - 11T^{2} \)
13 \( 1 + (-2.26 + 1.30i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.73 + 3.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.17 + 1.65i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 8.37iT - 23T^{2} \)
29 \( 1 + (5.16 + 5.16i)T + 29iT^{2} \)
31 \( 1 + (-6.16 + 6.16i)T - 31iT^{2} \)
41 \( 1 + (2.24 - 1.29i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.35iT - 43T^{2} \)
47 \( 1 + (0.101 - 0.101i)T - 47iT^{2} \)
53 \( 1 + (-2.60 + 0.698i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.77 - 10.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.11 + 1.37i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.55 + 9.52i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.81 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.21 - 3.21i)T - 73iT^{2} \)
79 \( 1 + (-13.7 - 3.67i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-15.6 + 4.18i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.75 - 1.00i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 - 4.12T + 97T^{2} \)
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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

−11.19356027718508600149226545548, −10.54382028956596153296391983357, −9.034330740556063883628338660958, −8.372665861622833539860047773320, −7.63946733808491022139817880664, −6.25015503757251874412868098956, −5.10552713398355235175083454336, −4.32804918716712579881828119945, −2.12270157727177786958389568602, −0.66396957496242465705181077443, 1.58573640071148988557311585511, 3.76549146108053812983290161026, 5.01948139303281826738770366146, 5.97725981087303421101390707439, 6.99471295298468195296414863955, 8.048939664158949765870130784234, 8.578801990199009122267945751117, 10.32552246035050252022791878623, 10.82323451091407373969747346668, 11.32271195560961727558232038160

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