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Properties

Label 2-605-55.8-c1-0-14
Degree $2$
Conductor $605$
Sign $0.155 - 0.987i$
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  + (−0.743 + 1.45i)3-s + (−1.90 − 0.618i)4-s + (2.22 − 0.238i)5-s + (0.186 + 0.256i)9-s + (2.31 − 2.31i)12-s + (−1.30 + 3.42i)15-s + (3.23 + 2.35i)16-s + (−4.37 − 0.919i)20-s + (6.15 + 6.15i)23-s + (4.88 − 1.06i)25-s + (−5.36 + 0.849i)27-s + (−8.04 + 5.84i)31-s + (−0.195 − 0.602i)36-s + (5.44 + 10.6i)37-s + (0.474 + 0.525i)45-s + ⋯
L(s)  = 1  + (−0.429 + 0.842i)3-s + (−0.951 − 0.309i)4-s + (0.994 − 0.106i)5-s + (0.0620 + 0.0853i)9-s + (0.668 − 0.668i)12-s + (−0.336 + 0.883i)15-s + (0.809 + 0.587i)16-s + (−0.978 − 0.205i)20-s + (1.28 + 1.28i)23-s + (0.977 − 0.212i)25-s + (−1.03 + 0.163i)27-s + (−1.44 + 1.05i)31-s + (−0.0326 − 0.100i)36-s + (0.894 + 1.75i)37-s + (0.0707 + 0.0782i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.878359 + 0.750863i\)
\(L(\frac12)\) \(\approx\) \(0.878359 + 0.750863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.238i)T \)
11 \( 1 \)
good2 \( 1 + (1.90 + 0.618i)T^{2} \)
3 \( 1 + (0.743 - 1.45i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.15 - 6.15i)T + 23iT^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.04 - 5.84i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.44 - 10.6i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (3.38 + 1.72i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-13.4 - 2.13i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (3.15 + 1.02i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.52 + 1.52i)T - 67iT^{2} \)
71 \( 1 + (-2.42 - 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (78.9 - 25.6i)T^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (-2.98 + 18.8i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.61313476653836170222584551903, −9.901181436883924235041379009811, −9.359573909264361961158972764905, −8.584150279858343721449815127763, −7.24437100954603116983393270227, −5.93949804546623387204700847743, −5.20824326164966529243115691919, −4.61264418342813598005421002602, −3.32686226208393163050055530465, −1.46614969890992294690247683445, 0.77592493464589688380579288666, 2.30680164289502968902825983905, 3.81790813532433645067113919949, 5.07007443540879725777741912779, 5.91076229218115290737933427256, 6.84072285208118634236425409807, 7.69507927543290154878792219008, 8.896493081161516870674850846857, 9.417970365712144377051299951609, 10.40394246329357589567828912012

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