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Properties

Label 2-380-20.7-c1-0-39
Degree $2$
Conductor $380$
Sign $0.259 + 0.965i$
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  + (0.912 + 1.08i)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 + (−2.43 + 1.43i)8-s + 0.592i·9-s + (−1.73 − 2.64i)10-s − 4.74i·11-s + (3.09 − 2.19i)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  + (0.645 + 0.764i)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 + (−0.861 + 0.507i)8-s + 0.197i·9-s + (−0.547 − 0.836i)10-s − 1.43i·11-s + (0.892 − 0.633i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.259 + 0.965i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.259 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708740 - 0.543185i\)
\(L(\frac12)\) \(\approx\) \(0.708740 - 0.543185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.912 - 1.08i)T \)
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

−11.29025583162901493185914660390, −10.82082671329194091808481763934, −8.719273274782989136718020472675, −8.190983136698962038864036760352, −7.24973150649372532647446231801, −6.27987760388826868613279707996, −5.61291719010583784797945174130, −4.26333474853128881415001582604, −3.23343037540657083998074101688, −0.53701488288438477331618364115, 1.98965668816097269078463139411, 3.91464338506096298883520301129, 4.33043980624931562322606461982, 5.39556628015679222193285184737, 6.49108742731726077294018320627, 7.79127129701910468142373971155, 9.165563081658358370455155752834, 10.03869539280773313122130803177, 10.88493661696901919861423149380, 11.61501520398900764524227711009

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