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Properties

Label 2-605-11.3-c1-0-7
Degree $2$
Conductor $605$
Sign $-0.999 + 0.0439i$
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  + (−0.335 + 0.243i)2-s + (0.874 + 2.68i)3-s + (−0.565 + 1.73i)4-s + (0.809 + 0.587i)5-s + (−0.947 − 0.688i)6-s + (0.618 − 1.90i)7-s + (−0.490 − 1.50i)8-s + (−4.04 + 2.93i)9-s − 0.414·10-s − 5.17·12-s + (−5.52 + 4.01i)13-s + (0.255 + 0.787i)14-s + (−0.874 + 2.68i)15-s + (−2.42 − 1.76i)16-s + (0.947 + 0.688i)17-s + (0.639 − 1.96i)18-s + ⋯
L(s)  = 1  + (−0.236 + 0.172i)2-s + (0.504 + 1.55i)3-s + (−0.282 + 0.869i)4-s + (0.361 + 0.262i)5-s + (−0.386 − 0.281i)6-s + (0.233 − 0.718i)7-s + (−0.173 − 0.533i)8-s + (−1.34 + 0.979i)9-s − 0.130·10-s − 1.49·12-s + (−1.53 + 1.11i)13-s + (0.0684 + 0.210i)14-s + (−0.225 + 0.694i)15-s + (−0.606 − 0.440i)16-s + (0.229 + 0.167i)17-s + (0.150 − 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.999 + 0.0439i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.999 + 0.0439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0270817 - 1.23159i\)
\(L(\frac12)\) \(\approx\) \(0.0270817 - 1.23159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.335 - 0.243i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.874 - 2.68i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.52 - 4.01i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.947 - 0.688i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + (2.36 - 7.28i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.13 + 3.47i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.85 + 5.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (0.874 + 2.68i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.277 - 0.201i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.98 - 9.18i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-10.7 - 7.82i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 + (-9.15 - 6.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.11 + 6.49i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.85 + 3.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.31T + 89T^{2} \)
97 \( 1 + (-6.19 + 4.50i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.76306715455917429781784602464, −10.08894492641809420161188859331, −9.240691250887631147336050190274, −8.865103221706561359651163065518, −7.57817649023577981735423921875, −6.96712848359239909196143264443, −5.22852002838099850340841017273, −4.38779298882035372044621552175, −3.66203863264971816897911900887, −2.53853777692249772949488075715, 0.68690710746519975681180053912, 2.01023181642491046215374526281, 2.73684152176539910130351854145, 4.95034246765207611352534874032, 5.69973230223964007165631226067, 6.62526006174658493359215718263, 7.74110393635709094760599492909, 8.325404912003998555298311763450, 9.367096083588972863567863761996, 9.937300359455188659031589767532

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