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Properties

Label 2-380-20.3-c1-0-51
Degree $2$
Conductor $380$
Sign $-0.564 + 0.825i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  + (1.08 − 0.912i)2-s + (1.34 − 1.34i)3-s + (0.335 − 1.97i)4-s + (−2.21 + 0.272i)5-s + (0.225 − 2.67i)6-s + (−0.697 − 0.697i)7-s + (−1.43 − 2.43i)8-s − 0.592i·9-s + (−2.14 + 2.31i)10-s − 4.74i·11-s + (−2.19 − 3.09i)12-s + (3.30 + 3.30i)13-s + (−1.39 − 0.117i)14-s + (−2.60 + 3.33i)15-s + (−3.77 − 1.32i)16-s + (−1.03 + 1.03i)17-s + ⋯
L(s)  = 1  + (0.764 − 0.645i)2-s + (0.773 − 0.773i)3-s + (0.167 − 0.985i)4-s + (−0.992 + 0.121i)5-s + (0.0920 − 1.09i)6-s + (−0.263 − 0.263i)7-s + (−0.507 − 0.861i)8-s − 0.197i·9-s + (−0.679 + 0.733i)10-s − 1.43i·11-s + (−0.633 − 0.892i)12-s + (0.916 + 0.916i)13-s + (−0.371 − 0.0313i)14-s + (−0.673 + 0.862i)15-s + (−0.943 − 0.330i)16-s + (−0.249 + 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.564 + 0.825i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.564 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.998164 - 1.89224i\)
\(L(\frac12)\) \(\approx\) \(0.998164 - 1.89224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 + 0.912i)T \)
5 \( 1 + (2.21 - 0.272i)T \)
19 \( 1 - T \)
good3 \( 1 + (-1.34 + 1.34i)T - 3iT^{2} \)
7 \( 1 + (0.697 + 0.697i)T + 7iT^{2} \)
11 \( 1 + 4.74iT - 11T^{2} \)
13 \( 1 + (-3.30 - 3.30i)T + 13iT^{2} \)
17 \( 1 + (1.03 - 1.03i)T - 17iT^{2} \)
23 \( 1 + (-6.40 + 6.40i)T - 23iT^{2} \)
29 \( 1 - 1.98iT - 29T^{2} \)
31 \( 1 - 9.29iT - 31T^{2} \)
37 \( 1 + (5.70 - 5.70i)T - 37iT^{2} \)
41 \( 1 - 9.41T + 41T^{2} \)
43 \( 1 + (1.89 - 1.89i)T - 43iT^{2} \)
47 \( 1 + (1.10 + 1.10i)T + 47iT^{2} \)
53 \( 1 + (0.00450 + 0.00450i)T + 53iT^{2} \)
59 \( 1 - 3.02T + 59T^{2} \)
61 \( 1 + 0.724T + 61T^{2} \)
67 \( 1 + (10.1 + 10.1i)T + 67iT^{2} \)
71 \( 1 - 9.29iT - 71T^{2} \)
73 \( 1 + (4.54 + 4.54i)T + 73iT^{2} \)
79 \( 1 + 0.319T + 79T^{2} \)
83 \( 1 + (-2.15 + 2.15i)T - 83iT^{2} \)
89 \( 1 - 8.13iT - 89T^{2} \)
97 \( 1 + (8.29 - 8.29i)T - 97iT^{2} \)
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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

−11.05160045468222067315581543662, −10.65644662635754137018227354485, −8.875381276949514661305089721384, −8.479410069318239158837292322067, −7.05570485938268095938516621156, −6.42519929370706486530778769975, −4.85406277715867640529219910156, −3.60142717864132236624264378653, −2.89796170666020682197711867817, −1.17951039138535198771281343662, 2.85388953043768187694844156477, 3.77821177728813309540648667748, 4.54464187355443901077600344348, 5.72639763710037849009277490969, 7.14825951131535712434626605266, 7.80360250759671102551827886752, 8.849744067609372937814505190644, 9.531922662473790509230248584914, 10.90343948171924302099391672680, 11.83483685590848252285652089764

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