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Properties

Label 2-372-372.191-c1-0-32
Degree $2$
Conductor $372$
Sign $0.717 + 0.696i$
Analytic cond. $2.97043$
Root an. cond. $1.72349$
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  + (−0.873 + 1.11i)2-s + (−0.703 − 1.58i)3-s + (−0.472 − 1.94i)4-s + (3.08 − 1.77i)5-s + (2.37 + 0.600i)6-s + (2.77 + 1.60i)7-s + (2.57 + 1.17i)8-s + (−2.00 + 2.22i)9-s + (−0.714 + 4.97i)10-s + (−3.05 − 5.29i)11-s + (−2.74 + 2.11i)12-s + (−0.185 − 0.321i)13-s + (−4.20 + 1.68i)14-s + (−4.98 − 3.62i)15-s + (−3.55 + 1.83i)16-s + (2.57 + 1.48i)17-s + ⋯
L(s)  = 1  + (−0.617 + 0.786i)2-s + (−0.406 − 0.913i)3-s + (−0.236 − 0.971i)4-s + (1.37 − 0.795i)5-s + (0.969 + 0.245i)6-s + (1.04 + 0.605i)7-s + (0.910 + 0.414i)8-s + (−0.669 + 0.742i)9-s + (−0.225 + 1.57i)10-s + (−0.921 − 1.59i)11-s + (−0.791 + 0.610i)12-s + (−0.0514 − 0.0891i)13-s + (−1.12 + 0.450i)14-s + (−1.28 − 0.935i)15-s + (−0.888 + 0.459i)16-s + (0.625 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(372\)    =    \(2^{2} \cdot 3 \cdot 31\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(2.97043\)
Root analytic conductor: \(1.72349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{372} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 372,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02756 - 0.417025i\)
\(L(\frac12)\) \(\approx\) \(1.02756 - 0.417025i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.873 - 1.11i)T \)
3 \( 1 + (0.703 + 1.58i)T \)
31 \( 1 + (2.68 - 4.87i)T \)
good5 \( 1 + (-3.08 + 1.77i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.77 - 1.60i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.05 + 5.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.185 + 0.321i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.57 - 1.48i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.49 - 2.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.65T + 23T^{2} \)
29 \( 1 + 9.84iT - 29T^{2} \)
37 \( 1 + (-0.414 + 0.718i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.17 + 1.83i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.72 + 0.993i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.401T + 47T^{2} \)
53 \( 1 + (3.45 - 1.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.44 - 2.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.54T + 61T^{2} \)
67 \( 1 + (-4.77 + 2.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.26 - 3.91i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.93 - 3.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.13 - 2.38i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.118 + 0.205i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−11.15375073651016657038810217592, −10.28035363446953603372174458841, −9.175615162006015902477347226875, −8.215328452178956589271573068307, −7.88058739681405362875652119284, −6.21410865423496120090201959809, −5.58970601561945880445612892333, −5.19890279194304309302443941653, −2.21208853569748929876698725460, −1.07069197533436288185423895793, 1.78931572707886740923799875819, 3.02830954861774926687370251497, 4.56833228248488994164276840125, 5.35142908801347644622456826276, 6.95776857670769031229418297837, 7.81393598601981498154844232762, 9.337951727801794166713908371371, 9.831628044428889590167972066285, 10.53455603040374158907308649190, 11.05508592155066671686781452427

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