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Properties

Label 2-185-185.43-c1-0-11
Degree $2$
Conductor $185$
Sign $-0.507 + 0.861i$
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  − 1.48i·2-s + (−2.07 + 2.07i)3-s − 0.193·4-s + (−1.48 − 1.67i)5-s + (3.07 + 3.07i)6-s + (1.07 − 1.07i)7-s − 2.67i·8-s − 5.63i·9-s + (−2.48 + 2.19i)10-s − 5.76i·11-s + (0.403 − 0.403i)12-s + 2.63i·13-s + (−1.59 − 1.59i)14-s + (6.55 + 0.403i)15-s − 4.35·16-s − 3.28·17-s + ⋯
L(s)  = 1  − 1.04i·2-s + (−1.19 + 1.19i)3-s − 0.0969·4-s + (−0.662 − 0.749i)5-s + (1.25 + 1.25i)6-s + (0.407 − 0.407i)7-s − 0.945i·8-s − 1.87i·9-s + (−0.784 + 0.693i)10-s − 1.73i·11-s + (0.116 − 0.116i)12-s + 0.731i·13-s + (−0.426 − 0.426i)14-s + (1.69 + 0.104i)15-s − 1.08·16-s − 0.797·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.507 + 0.861i$
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: \((2,\ 185,\ (\ :1/2),\ -0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350935 - 0.614038i\)
\(L(\frac12)\) \(\approx\) \(0.350935 - 0.614038i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.48 + 1.67i)T \)
37 \( 1 + (1 + 6i)T \)
good2 \( 1 + 1.48iT - 2T^{2} \)
3 \( 1 + (2.07 - 2.07i)T - 3iT^{2} \)
7 \( 1 + (-1.07 + 1.07i)T - 7iT^{2} \)
11 \( 1 + 5.76iT - 11T^{2} \)
13 \( 1 - 2.63iT - 13T^{2} \)
17 \( 1 + 3.28T + 17T^{2} \)
19 \( 1 + (-4.07 + 4.07i)T - 19iT^{2} \)
23 \( 1 - 1.76iT - 23T^{2} \)
29 \( 1 + (-2.35 - 2.35i)T + 29iT^{2} \)
31 \( 1 + (1.46 - 1.46i)T - 31iT^{2} \)
41 \( 1 - 9.38iT - 41T^{2} \)
43 \( 1 + 8.73iT - 43T^{2} \)
47 \( 1 + (-1.94 + 1.94i)T - 47iT^{2} \)
53 \( 1 + (0.518 + 0.518i)T + 53iT^{2} \)
59 \( 1 + (-5.46 + 5.46i)T - 59iT^{2} \)
61 \( 1 + (-4.19 + 4.19i)T - 61iT^{2} \)
67 \( 1 + (-10.7 - 10.7i)T + 67iT^{2} \)
71 \( 1 - 4.57T + 71T^{2} \)
73 \( 1 + (1.16 - 1.16i)T - 73iT^{2} \)
79 \( 1 + (7.55 - 7.55i)T - 79iT^{2} \)
83 \( 1 + (0.559 + 0.559i)T + 83iT^{2} \)
89 \( 1 + (6.02 + 6.02i)T + 89iT^{2} \)
97 \( 1 + 4.15T + 97T^{2} \)
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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

−11.59895467921061331503256392103, −11.38314260710380896091400597748, −10.78761887917138072550409246414, −9.582972605061652892973776429325, −8.693959132063742600281674580644, −6.89554731764850968695159465336, −5.47302170901219952041803599685, −4.39431921498008810296840134600, −3.48787935946983672189902215610, −0.74612836214952433856787255915, 2.17047087426007556117523861101, 4.82189375207204631197551845640, 5.89267320017335200137556511034, 6.84864841944309190451424492434, 7.43424036496168895164331973710, 8.181318439069842816258176089207, 10.22320625152695695677192940662, 11.31421560810015575576035693507, 11.94853486010976761642751970302, 12.69315054629233710570648590582

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