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Properties

Label 2-605-55.7-c1-0-30
Degree $2$
Conductor $605$
Sign $0.288 + 0.957i$
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  + (0.568 − 0.0900i)2-s + (0.349 + 0.685i)3-s + (−1.58 + 0.515i)4-s + (−1.84 − 1.26i)5-s + (0.260 + 0.358i)6-s + (1.06 + 0.541i)7-s + (−1.88 + 0.959i)8-s + (1.41 − 1.94i)9-s + (−1.16 − 0.550i)10-s + (−0.908 − 0.908i)12-s + (−0.950 − 6.00i)13-s + (0.653 + 0.212i)14-s + (0.219 − 1.70i)15-s + (1.71 − 1.24i)16-s + (0.0580 − 0.366i)17-s + (0.629 − 1.23i)18-s + ⋯
L(s)  = 1  + (0.402 − 0.0636i)2-s + (0.201 + 0.395i)3-s + (−0.793 + 0.257i)4-s + (−0.825 − 0.563i)5-s + (0.106 + 0.146i)6-s + (0.401 + 0.204i)7-s + (−0.665 + 0.339i)8-s + (0.471 − 0.649i)9-s + (−0.368 − 0.174i)10-s + (−0.262 − 0.262i)12-s + (−0.263 − 1.66i)13-s + (0.174 + 0.0567i)14-s + (0.0566 − 0.440i)15-s + (0.428 − 0.311i)16-s + (0.0140 − 0.0888i)17-s + (0.148 − 0.291i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (282, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.940492 - 0.699065i\)
\(L(\frac12)\) \(\approx\) \(0.940492 - 0.699065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.84 + 1.26i)T \)
11 \( 1 \)
good2 \( 1 + (-0.568 + 0.0900i)T + (1.90 - 0.618i)T^{2} \)
3 \( 1 + (-0.349 - 0.685i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-1.06 - 0.541i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (0.950 + 6.00i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.0580 + 0.366i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-0.425 + 1.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.48 + 3.48i)T - 23iT^{2} \)
29 \( 1 + (1.89 + 5.82i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.00 - 1.45i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.35 - 2.65i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (5.82 + 1.89i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (6.75 + 6.75i)T + 43iT^{2} \)
47 \( 1 + (-1.13 + 0.579i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.402 + 0.0638i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (0.178 - 0.0580i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.401 - 0.552i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-7.14 - 7.14i)T + 67iT^{2} \)
71 \( 1 + (0.836 - 0.607i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.07 - 4.07i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-9.41 - 6.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.2 + 1.78i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 - 8.04iT - 89T^{2} \)
97 \( 1 + (1.32 + 8.33i)T + (-92.2 + 29.9i)T^{2} \)
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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

−10.35900643966273461153361171822, −9.556916991082261970513974578001, −8.537917787977533019028320315578, −8.193668744191061972513930712350, −7.00367208042113632132887907414, −5.44294589701674740251190226685, −4.81163261200109377668835346343, −3.87157529174795731166187574423, −3.02439764933648337470693349899, −0.61102765441061384731892006590, 1.62802056959361350478873025329, 3.31186509554263465410892517012, 4.36074260652699892630887743175, 5.00362549108634292739695944438, 6.48838024601604191666003209658, 7.25246205295912762688821162220, 8.098427270251995755743774907792, 9.022802659137561803171530038910, 9.950196185024598832124131117695, 10.94493993894886929778456421531

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