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Properties

Label 2-605-121.16-c1-0-9
Degree $2$
Conductor $605$
Sign $-0.329 - 0.944i$
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  + (0.252 + 0.0289i)2-s + (2.55 + 1.85i)3-s + (−1.88 − 0.438i)4-s + (−0.564 + 0.825i)5-s + (0.590 + 0.541i)6-s + (−0.840 − 0.864i)7-s + (−0.941 − 0.336i)8-s + (2.14 + 6.60i)9-s + (−0.166 + 0.192i)10-s + (2.57 + 2.09i)11-s + (−3.99 − 4.61i)12-s + (−2.11 + 3.50i)13-s + (−0.187 − 0.242i)14-s + (−2.97 + 1.05i)15-s + (3.24 + 1.59i)16-s + (−2.71 + 2.21i)17-s + ⋯
L(s)  = 1  + (0.178 + 0.0204i)2-s + (1.47 + 1.07i)3-s + (−0.942 − 0.219i)4-s + (−0.252 + 0.369i)5-s + (0.241 + 0.221i)6-s + (−0.317 − 0.326i)7-s + (−0.333 − 0.118i)8-s + (0.715 + 2.20i)9-s + (−0.0526 + 0.0607i)10-s + (0.775 + 0.630i)11-s + (−1.15 − 1.33i)12-s + (−0.585 + 0.971i)13-s + (−0.0500 − 0.0648i)14-s + (−0.766 + 0.273i)15-s + (0.811 + 0.398i)16-s + (−0.658 + 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.329 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05654 + 1.48737i\)
\(L(\frac12)\) \(\approx\) \(1.05654 + 1.48737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.564 - 0.825i)T \)
11 \( 1 + (-2.57 - 2.09i)T \)
good2 \( 1 + (-0.252 - 0.0289i)T + (1.94 + 0.452i)T^{2} \)
3 \( 1 + (-2.55 - 1.85i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.840 + 0.864i)T + (-0.199 + 6.99i)T^{2} \)
13 \( 1 + (2.11 - 3.50i)T + (-6.06 - 11.4i)T^{2} \)
17 \( 1 + (2.71 - 2.21i)T + (3.37 - 16.6i)T^{2} \)
19 \( 1 + (1.09 + 0.0627i)T + (18.8 + 2.16i)T^{2} \)
23 \( 1 + (-0.207 + 1.44i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + (0.00387 - 0.00995i)T + (-21.3 - 19.6i)T^{2} \)
31 \( 1 + (-7.98 - 3.37i)T + (21.6 + 22.2i)T^{2} \)
37 \( 1 + (-0.718 + 8.36i)T + (-36.4 - 6.30i)T^{2} \)
41 \( 1 + (10.0 - 5.66i)T + (21.1 - 35.1i)T^{2} \)
43 \( 1 + (-8.91 + 2.61i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-0.789 + 3.89i)T + (-43.2 - 18.2i)T^{2} \)
53 \( 1 + (-6.26 + 3.07i)T + (32.3 - 41.9i)T^{2} \)
59 \( 1 + (-10.9 - 6.18i)T + (30.4 + 50.5i)T^{2} \)
61 \( 1 + (8.31 - 0.953i)T + (59.4 - 13.8i)T^{2} \)
67 \( 1 + (-0.706 + 1.54i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (0.458 + 1.74i)T + (-61.8 + 34.9i)T^{2} \)
73 \( 1 + (3.11 + 5.90i)T + (-41.2 + 60.2i)T^{2} \)
79 \( 1 + (0.247 + 8.65i)T + (-78.8 + 4.51i)T^{2} \)
83 \( 1 + (-6.51 - 1.12i)T + (78.1 + 27.8i)T^{2} \)
89 \( 1 + (2.44 + 1.57i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-1.94 - 2.84i)T + (-35.1 + 90.3i)T^{2} \)
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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

−10.30860148298059561645427389579, −10.07511044339543899839005476130, −8.972480919355194053561867831939, −8.794117932296284460512550145840, −7.56674041818590595898944648725, −6.52927439120258081746519327777, −4.82098442367593653900675329909, −4.20489040891560117418111334751, −3.56208841419267601657251087832, −2.19949375707398776220190905189, 0.862836707606718212779907464806, 2.62822281957571339136156611215, 3.43558214861132710941757410744, 4.52159822825097556883240539848, 5.93286723419917164697861850600, 7.04246599250899296774151275470, 8.029761870713497449941261292585, 8.538529046640418367870939631796, 9.208551700646292304594277174624, 9.934195777214992060398496305631

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