Properties

Label 2-605-55.13-c1-0-25
Degree $2$
Conductor $605$
Sign $-0.191 - 0.981i$
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  + (−1.10 + 0.562i)2-s + (2.51 + 0.397i)3-s + (−0.274 + 0.377i)4-s + (1.89 + 1.18i)5-s + (−2.99 + 0.973i)6-s + (0.542 + 3.42i)7-s + (0.477 − 3.01i)8-s + (3.30 + 1.07i)9-s + (−2.75 − 0.235i)10-s + (−0.839 + 0.839i)12-s + (0.335 + 0.658i)13-s + (−2.52 − 3.47i)14-s + (4.30 + 3.72i)15-s + (0.880 + 2.70i)16-s + (−0.280 + 0.550i)17-s + (−4.24 + 0.672i)18-s + ⋯
L(s)  = 1  + (−0.780 + 0.397i)2-s + (1.45 + 0.229i)3-s + (−0.137 + 0.188i)4-s + (0.849 + 0.528i)5-s + (−1.22 + 0.397i)6-s + (0.205 + 1.29i)7-s + (0.168 − 1.06i)8-s + (1.10 + 0.357i)9-s + (−0.872 − 0.0744i)10-s + (−0.242 + 0.242i)12-s + (0.0931 + 0.182i)13-s + (−0.674 − 0.928i)14-s + (1.11 + 0.961i)15-s + (0.220 + 0.677i)16-s + (−0.0679 + 0.133i)17-s + (−1.00 + 0.158i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.191 - 0.981i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.191 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08738 + 1.31966i\)
\(L(\frac12)\) \(\approx\) \(1.08738 + 1.31966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.89 - 1.18i)T \)
11 \( 1 \)
good2 \( 1 + (1.10 - 0.562i)T + (1.17 - 1.61i)T^{2} \)
3 \( 1 + (-2.51 - 0.397i)T + (2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.542 - 3.42i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (-0.335 - 0.658i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.280 - 0.550i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-2.54 + 1.84i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.30 + 4.30i)T + 23iT^{2} \)
29 \( 1 + (2.34 + 1.70i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.937 + 2.88i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.27 - 0.359i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (0.599 + 0.825i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-4.07 + 4.07i)T - 43iT^{2} \)
47 \( 1 + (0.169 - 1.07i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.82 + 1.94i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-4.35 + 5.98i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.56 - 1.15i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.39 - 9.39i)T - 67iT^{2} \)
71 \( 1 + (-1.11 - 3.43i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.531 + 0.0841i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (3.77 - 11.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-14.2 - 7.27i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + (-6.51 - 12.7i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.37850815893916490863346350963, −9.609097827854337169202894321662, −9.058018979391978088124654028516, −8.491562625787402536571734778303, −7.69936190375829942427883798249, −6.67251815710414854900548721337, −5.58554188838792103298984949098, −4.06990827034634647612789295506, −2.89863904435111935086807198403, −2.06123414033693352169038970998, 1.16984783509256012993508119560, 2.03488893805031809607596038435, 3.41746538681857087857024097862, 4.65092142604365808522823292644, 5.84082249917012869196865529917, 7.35940698222552074771985849295, 7.973577397061231620954338560659, 8.834358170710144026382244896211, 9.484374491869764381386222051910, 10.11859649661438170910457386441

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