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Properties

Label 2-605-11.5-c1-0-20
Degree $2$
Conductor $605$
Sign $0.955 - 0.296i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (0.810 + 2.49i)2-s + (−1.08 − 0.787i)3-s + (−3.95 + 2.87i)4-s + (0.309 − 0.951i)5-s + (1.08 − 3.34i)6-s + (2.12 − 1.54i)7-s + (−6.12 − 4.44i)8-s + (−0.372 − 1.14i)9-s + 2.62·10-s + 6.54·12-s + (−2.07 − 6.39i)13-s + (5.56 + 4.04i)14-s + (−1.08 + 0.787i)15-s + (3.11 − 9.59i)16-s + (0.551 − 1.69i)17-s + (2.55 − 1.85i)18-s + ⋯
L(s)  = 1  + (0.573 + 1.76i)2-s + (−0.625 − 0.454i)3-s + (−1.97 + 1.43i)4-s + (0.138 − 0.425i)5-s + (0.443 − 1.36i)6-s + (0.802 − 0.582i)7-s + (−2.16 − 1.57i)8-s + (−0.124 − 0.381i)9-s + 0.829·10-s + 1.88·12-s + (−0.576 − 1.77i)13-s + (1.48 + 1.08i)14-s + (−0.279 + 0.203i)15-s + (0.779 − 2.39i)16-s + (0.133 − 0.411i)17-s + (0.602 − 0.437i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.955 - 0.296i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.955 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25330 + 0.190123i\)
\(L(\frac12)\) \(\approx\) \(1.25330 + 0.190123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.810 - 2.49i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.08 + 0.787i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.12 + 1.54i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.07 + 6.39i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.551 + 1.69i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.19 - 0.870i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.20T + 23T^{2} \)
29 \( 1 + (0.952 - 0.692i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.41 + 4.34i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.549 + 0.399i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.44 + 3.23i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.80T + 43T^{2} \)
47 \( 1 + (5.38 + 3.91i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.41 - 13.5i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.14 + 2.28i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.992 + 3.05i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.66T + 67T^{2} \)
71 \( 1 + (-1.20 + 3.69i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.88 + 2.09i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.67 + 14.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.26 + 6.97i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.44T + 89T^{2} \)
97 \( 1 + (-0.455 - 1.40i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.73249835126520707261516052794, −9.490981772305911977198187144596, −8.523213799616271783509419236411, −7.62083834289964308941899107106, −7.21558943793519023933783132619, −6.03983382956445595732043151947, −5.37107420898288409484013834568, −4.72807978645272201896196290518, −3.39231997076696283532888515494, −0.66667153995845537943967269804, 1.69221397382170508039801255770, 2.59308801211833292817141691976, 3.96400315155914246184678773459, 4.90037475487086683341460540997, 5.38507872758877775288167957066, 6.77367254940699683214560480592, 8.404810196263231453504198367659, 9.351264393662475524629005643209, 10.04598633790604015354047198740, 10.96666645412049227031823546304

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