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Properties

Label 2-605-55.43-c1-0-30
Degree $2$
Conductor $605$
Sign $0.971 + 0.237i$
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  + (1.31 + 1.31i)2-s + (−1.09 − 1.09i)3-s + 1.44i·4-s + (2.21 + 0.283i)5-s − 2.86i·6-s + (−1.45 − 1.45i)7-s + (0.725 − 0.725i)8-s − 0.617i·9-s + (2.54 + 3.28i)10-s + (1.57 − 1.57i)12-s + (2.13 − 2.13i)13-s − 3.83i·14-s + (−2.11 − 2.73i)15-s + 4.80·16-s + (−4.86 − 4.86i)17-s + (0.810 − 0.810i)18-s + ⋯
L(s)  = 1  + (0.928 + 0.928i)2-s + (−0.630 − 0.630i)3-s + 0.723i·4-s + (0.991 + 0.126i)5-s − 1.17i·6-s + (−0.551 − 0.551i)7-s + (0.256 − 0.256i)8-s − 0.205i·9-s + (0.803 + 1.03i)10-s + (0.455 − 0.455i)12-s + (0.590 − 0.590i)13-s − 1.02i·14-s + (−0.545 − 0.704i)15-s + 1.20·16-s + (−1.18 − 1.18i)17-s + (0.190 − 0.190i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17634 - 0.262400i\)
\(L(\frac12)\) \(\approx\) \(2.17634 - 0.262400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.21 - 0.283i)T \)
11 \( 1 \)
good2 \( 1 + (-1.31 - 1.31i)T + 2iT^{2} \)
3 \( 1 + (1.09 + 1.09i)T + 3iT^{2} \)
7 \( 1 + (1.45 + 1.45i)T + 7iT^{2} \)
13 \( 1 + (-2.13 + 2.13i)T - 13iT^{2} \)
17 \( 1 + (4.86 + 4.86i)T + 17iT^{2} \)
19 \( 1 + 4.07T + 19T^{2} \)
23 \( 1 + (-5.40 - 5.40i)T + 23iT^{2} \)
29 \( 1 - 5.13T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (-3.38 + 3.38i)T - 37iT^{2} \)
41 \( 1 - 7.55iT - 41T^{2} \)
43 \( 1 + (-2.58 + 2.58i)T - 43iT^{2} \)
47 \( 1 + (-0.313 + 0.313i)T - 47iT^{2} \)
53 \( 1 + (5.18 + 5.18i)T + 53iT^{2} \)
59 \( 1 - 12.9iT - 59T^{2} \)
61 \( 1 - 5.35iT - 61T^{2} \)
67 \( 1 + (3.40 - 3.40i)T - 67iT^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + (6.35 - 6.35i)T - 73iT^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (-0.789 + 0.789i)T - 83iT^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + (-5.96 + 5.96i)T - 97iT^{2} \)
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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

−10.67739951855571435141432995701, −9.798990312752229974255017741893, −8.809462465901043581808967565095, −7.31567714229213265133555377873, −6.75848169413520435374391822614, −6.16445417362855777410963670396, −5.41052231321532341563135869746, −4.35667287316234836482786623214, −2.97382446851921637534795647323, −1.06832357382458605072370777604, 1.88324336037111510339235161980, 2.83154340635327336477932170493, 4.30982716818544257367369247524, 4.79751699434030591062027534409, 5.99706055102741990895584872361, 6.46191038121294292295574044061, 8.391884850711526060522542193137, 9.149272986038950436170140661637, 10.34007783342809847091815182508, 10.70386066795330762842955112532

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