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Properties

Label 2-605-55.43-c1-0-37
Degree $2$
Conductor $605$
Sign $-0.619 + 0.785i$
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  + (−0.187 − 0.187i)2-s + (−0.0374 − 0.0374i)3-s − 1.92i·4-s + (1.76 + 1.37i)5-s + 0.0140i·6-s + (−2.30 − 2.30i)7-s + (−0.735 + 0.735i)8-s − 2.99i·9-s + (−0.0717 − 0.587i)10-s + (−0.0722 + 0.0722i)12-s + (−1.59 + 1.59i)13-s + 0.862i·14-s + (−0.0143 − 0.117i)15-s − 3.58·16-s + (−4.70 − 4.70i)17-s + (−0.560 + 0.560i)18-s + ⋯
L(s)  = 1  + (−0.132 − 0.132i)2-s + (−0.0216 − 0.0216i)3-s − 0.964i·4-s + (0.787 + 0.616i)5-s + 0.00572i·6-s + (−0.870 − 0.870i)7-s + (−0.260 + 0.260i)8-s − 0.999i·9-s + (−0.0226 − 0.185i)10-s + (−0.0208 + 0.0208i)12-s + (−0.442 + 0.442i)13-s + 0.230i·14-s + (−0.00370 − 0.0303i)15-s − 0.896·16-s + (−1.14 − 1.14i)17-s + (−0.132 + 0.132i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.619 + 0.785i$
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: \((2,\ 605,\ (\ :1/2),\ -0.619 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465618 - 0.960381i\)
\(L(\frac12)\) \(\approx\) \(0.465618 - 0.960381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.76 - 1.37i)T \)
11 \( 1 \)
good2 \( 1 + (0.187 + 0.187i)T + 2iT^{2} \)
3 \( 1 + (0.0374 + 0.0374i)T + 3iT^{2} \)
7 \( 1 + (2.30 + 2.30i)T + 7iT^{2} \)
13 \( 1 + (1.59 - 1.59i)T - 13iT^{2} \)
17 \( 1 + (4.70 + 4.70i)T + 17iT^{2} \)
19 \( 1 - 6.13T + 19T^{2} \)
23 \( 1 + (2.64 + 2.64i)T + 23iT^{2} \)
29 \( 1 + 5.84T + 29T^{2} \)
31 \( 1 - 3.89T + 31T^{2} \)
37 \( 1 + (-4.56 + 4.56i)T - 37iT^{2} \)
41 \( 1 + 1.80iT - 41T^{2} \)
43 \( 1 + (1.54 - 1.54i)T - 43iT^{2} \)
47 \( 1 + (-2.48 + 2.48i)T - 47iT^{2} \)
53 \( 1 + (6.60 + 6.60i)T + 53iT^{2} \)
59 \( 1 + 4.11iT - 59T^{2} \)
61 \( 1 + 1.77iT - 61T^{2} \)
67 \( 1 + (8.68 - 8.68i)T - 67iT^{2} \)
71 \( 1 - 6.89T + 71T^{2} \)
73 \( 1 + (-5.99 + 5.99i)T - 73iT^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + (-7.44 + 7.44i)T - 83iT^{2} \)
89 \( 1 - 2.26iT - 89T^{2} \)
97 \( 1 + (4.75 - 4.75i)T - 97iT^{2} \)
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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

−10.13117655948072462911215019045, −9.545793085200693055812004804618, −9.188616351056107639611904832205, −7.29171354059820258304216150413, −6.67367800908579113563392949481, −6.00820056468988755490249331287, −4.83268007883549837759317708881, −3.46706713802752633014074834882, −2.23963350004156273755214254994, −0.59044705092185285739759652182, 2.09216016402993899264028291565, 3.08606032677499085332102397952, 4.49773958741662255699363530775, 5.57313238529470869235592991507, 6.37747982572135529343358704405, 7.61815352293700758403520527211, 8.339499547384990713054578234098, 9.253908750741844556236639755005, 9.790304340870529507056941005525, 10.96216170848438699262087804492

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