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Properties

Label 2-185-185.68-c1-0-7
Degree $2$
Conductor $185$
Sign $-0.309 + 0.950i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−2 + 2i)3-s − 4-s + (−1 + 2i)5-s + (2 − 2i)6-s + 3·8-s − 5i·9-s + (1 − 2i)10-s − 4i·11-s + (2 − 2i)12-s − 4·13-s + (−2 − 6i)15-s − 16-s + 2i·17-s + 5i·18-s + ⋯
L(s)  = 1  − 0.707·2-s + (−1.15 + 1.15i)3-s − 0.5·4-s + (−0.447 + 0.894i)5-s + (0.816 − 0.816i)6-s + 1.06·8-s − 1.66i·9-s + (0.316 − 0.632i)10-s − 1.20i·11-s + (0.577 − 0.577i)12-s − 1.10·13-s + (−0.516 − 1.54i)15-s − 0.250·16-s + 0.485i·17-s + 1.17i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 - 2i)T \)
37 \( 1 + (1 - 6i)T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + (2 - 2i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (1 - i)T - 29iT^{2} \)
31 \( 1 + (6 + 6i)T + 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-8 + 8i)T - 47iT^{2} \)
53 \( 1 + (9 + 9i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 + (6 + 6i)T + 67iT^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (11 - 11i)T - 73iT^{2} \)
79 \( 1 + (6 + 6i)T + 79iT^{2} \)
83 \( 1 + (2 + 2i)T + 83iT^{2} \)
89 \( 1 + (1 - i)T - 89iT^{2} \)
97 \( 1 + 4iT - 97T^{2} \)
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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

−11.79925645971910819215343496929, −11.05233437961054836011496367652, −10.36069300317884076015181233277, −9.620012553799449594182753328326, −8.459038134695047937455211668892, −7.16788345206229107744699076591, −5.80487837787603637191963573195, −4.69954442347950675612313087224, −3.51600518630916943386036690955, 0, 1.56723760771432894279445474181, 4.62713613916430467161249897649, 5.33469939284089197070161284126, 7.14977691777917398809994079094, 7.53375307058247587357249366422, 8.867023978556716664989436402135, 9.842374305639781560858925177247, 11.04441630478008108341163433690, 12.16863657551990597683003049864

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