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Properties

Label 2-185-185.43-c1-0-2
Degree $2$
Conductor $185$
Sign $-0.887 - 0.461i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + i·2-s + (−1 + i)3-s + 4-s + (−2 − i)5-s + (−1 − i)6-s + (−3 + 3i)7-s + 3i·8-s + i·9-s + (1 − 2i)10-s − 2i·11-s + (−1 + i)12-s + 2i·13-s + (−3 − 3i)14-s + (3 − i)15-s − 16-s + 4·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + (−0.408 − 0.408i)6-s + (−1.13 + 1.13i)7-s + 1.06i·8-s + 0.333i·9-s + (0.316 − 0.632i)10-s − 0.603i·11-s + (−0.288 + 0.288i)12-s + 0.554i·13-s + (−0.801 − 0.801i)14-s + (0.774 − 0.258i)15-s − 0.250·16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.887 - 0.461i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186275 + 0.761936i\)
\(L(\frac12)\) \(\approx\) \(0.186275 + 0.761936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
37 \( 1 + (-6 - i)T \)
good2 \( 1 - iT - 2T^{2} \)
3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + (-5 + 5i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 + (7 - 7i)T - 59iT^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + (3 - 3i)T - 79iT^{2} \)
83 \( 1 + (5 + 5i)T + 83iT^{2} \)
89 \( 1 + (5 + 5i)T + 89iT^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−12.68622784951561515177893531225, −12.04471604417061927192935491645, −11.15856402620335126794905674713, −10.13784840830060668236522434062, −8.755790042292369064489530030519, −8.006684532421444278784206510283, −6.53218383237364485054161855706, −5.79875921088286720469674759935, −4.60608325564846972986619446651, −2.89122140575968583395944552281, 0.75401270737500311514487533493, 3.04597537633662380246768635930, 4.00693683997329616539485261832, 6.18992396753207268461596818254, 7.01375535963790216563019695725, 7.64221805809677080574720146986, 9.685110375655533509715451054863, 10.40888189373920789067404749174, 11.32049532116622859071951103366, 12.19097081145056866600852743597

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