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Properties

Label 2-605-55.49-c1-0-13
Degree $2$
Conductor $605$
Sign $-0.605 - 0.795i$
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  + (−1.26 + 0.410i)2-s + (−0.190 + 0.138i)4-s + (−0.690 + 2.12i)5-s + (2.52 + 3.47i)7-s + (1.74 − 2.40i)8-s + (0.927 + 2.85i)9-s − 2.96i·10-s + (6.61 − 2.14i)13-s + (−4.61 − 3.35i)14-s + (−1.07 + 3.30i)16-s + (−1.56 − 0.507i)17-s + (−2.34 − 3.22i)18-s + (−0.163 − 0.502i)20-s + (−4.04 − 2.93i)25-s + (−7.47 + 5.42i)26-s + ⋯
L(s)  = 1  + (−0.893 + 0.290i)2-s + (−0.0954 + 0.0693i)4-s + (−0.309 + 0.951i)5-s + (0.954 + 1.31i)7-s + (0.617 − 0.849i)8-s + (0.309 + 0.951i)9-s − 0.939i·10-s + (1.83 − 0.596i)13-s + (−1.23 − 0.896i)14-s + (−0.268 + 0.825i)16-s + (−0.378 − 0.123i)17-s + (−0.552 − 0.759i)18-s + (−0.0364 − 0.112i)20-s + (−0.809 − 0.587i)25-s + (−1.46 + 1.06i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410450 + 0.827929i\)
\(L(\frac12)\) \(\approx\) \(0.410450 + 0.827929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.690 - 2.12i)T \)
11 \( 1 \)
good2 \( 1 + (1.26 - 0.410i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-2.52 - 3.47i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-6.61 + 2.14i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.56 + 0.507i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.76 - 8.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.23 + 2.35i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-2.47 + 7.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.21 + 9.92i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (17.3 + 5.62i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (78.4 - 57.0i)T^{2} \)
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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

−10.84037822138629910885660845504, −10.16412555297149999179561351013, −8.851102265527746603573021813197, −8.388742512207712785808732297052, −7.73390275122540994605841394421, −6.69775636375154975185789269537, −5.62349326314226255484840572850, −4.44511932053291882400995129062, −3.12475276566318883998019062865, −1.68066974952694402511355172313, 0.827110009784552620750718314605, 1.52191771568781573467269811470, 4.01821926490180502078822441261, 4.36051420484588114191276144499, 5.75624345990572044144960985812, 7.03053854039136896605090045220, 8.095329096616609466933830085559, 8.586576585793625998736418929510, 9.412829174572902199396053191421, 10.24610463802248821555635607373

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