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Properties

Label 2-1860-155.123-c1-0-15
Degree $2$
Conductor $1860$
Sign $0.577 - 0.816i$
Analytic cond. $14.8521$
Root an. cond. $3.85385$
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  + (−0.707 + 0.707i)3-s + (1.94 + 1.10i)5-s + (−1.30 + 1.30i)7-s − 1.00i·9-s − 0.348i·11-s + (3.81 − 3.81i)13-s + (−2.15 + 0.593i)15-s + (0.867 + 0.867i)17-s + 3.13i·19-s − 1.84i·21-s + (3.18 − 3.18i)23-s + (2.56 + 4.29i)25-s + (0.707 + 0.707i)27-s + 3.05·29-s + (1.80 − 5.26i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.869 + 0.493i)5-s + (−0.492 + 0.492i)7-s − 0.333i·9-s − 0.105i·11-s + (1.05 − 1.05i)13-s + (−0.556 + 0.153i)15-s + (0.210 + 0.210i)17-s + 0.719i·19-s − 0.401i·21-s + (0.665 − 0.665i)23-s + (0.512 + 0.858i)25-s + (0.136 + 0.136i)27-s + 0.567·29-s + (0.324 − 0.945i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.793629202\)
\(L(\frac12)\) \(\approx\) \(1.793629202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.94 - 1.10i)T \)
31 \( 1 + (-1.80 + 5.26i)T \)
good7 \( 1 + (1.30 - 1.30i)T - 7iT^{2} \)
11 \( 1 + 0.348iT - 11T^{2} \)
13 \( 1 + (-3.81 + 3.81i)T - 13iT^{2} \)
17 \( 1 + (-0.867 - 0.867i)T + 17iT^{2} \)
19 \( 1 - 3.13iT - 19T^{2} \)
23 \( 1 + (-3.18 + 3.18i)T - 23iT^{2} \)
29 \( 1 - 3.05T + 29T^{2} \)
37 \( 1 + (-2.25 - 2.25i)T + 37iT^{2} \)
41 \( 1 - 8.13T + 41T^{2} \)
43 \( 1 + (1.78 - 1.78i)T - 43iT^{2} \)
47 \( 1 + (3.54 - 3.54i)T - 47iT^{2} \)
53 \( 1 + (-1.20 + 1.20i)T - 53iT^{2} \)
59 \( 1 - 3.43iT - 59T^{2} \)
61 \( 1 - 0.764iT - 61T^{2} \)
67 \( 1 + (0.999 - 0.999i)T - 67iT^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + (6.15 - 6.15i)T - 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + (5.99 - 5.99i)T - 83iT^{2} \)
89 \( 1 + 7.46T + 89T^{2} \)
97 \( 1 + (1.18 - 1.18i)T - 97iT^{2} \)
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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

−9.513041479092035754834129168205, −8.684065339151511657734590416539, −7.87162306914518767316815023457, −6.69883733001898407201916093058, −5.93685620079483690070856077222, −5.70121565756047196744068060160, −4.47395973036438532358461157075, −3.32851896799910233038162204287, −2.62243602469403009868852085731, −1.10860325091580344686642328189, 0.870010788034445359425186103928, 1.83557862769761041296972669653, 3.10439223082487746724286723034, 4.29839918283359721560754505076, 5.12460400049343773585387978725, 6.04368121181628214108772502044, 6.66407945682369257402432595950, 7.31764292863082892481483798890, 8.512445009423112227672723570967, 9.116703793307833773177300455281

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