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Properties

Label 2-605-55.18-c1-0-42
Degree $2$
Conductor $605$
Sign $-0.986 - 0.166i$
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  + (−0.391 − 2.47i)2-s + (2.33 − 1.19i)3-s + (−4.04 + 1.31i)4-s + (2.02 − 0.958i)5-s + (−3.85 − 5.30i)6-s + (0.167 − 0.328i)7-s + (2.56 + 5.02i)8-s + (2.28 − 3.14i)9-s + (−3.15 − 4.61i)10-s + (−7.89 + 7.89i)12-s + (−4.37 + 0.693i)13-s + (−0.877 − 0.284i)14-s + (3.58 − 4.64i)15-s + (4.52 − 3.29i)16-s + (0.849 + 0.134i)17-s + (−8.66 − 4.41i)18-s + ⋯
L(s)  = 1  + (−0.276 − 1.74i)2-s + (1.34 − 0.687i)3-s + (−2.02 + 0.657i)4-s + (0.903 − 0.428i)5-s + (−1.57 − 2.16i)6-s + (0.0632 − 0.124i)7-s + (0.905 + 1.77i)8-s + (0.761 − 1.04i)9-s + (−0.998 − 1.45i)10-s + (−2.27 + 2.27i)12-s + (−1.21 + 0.192i)13-s + (−0.234 − 0.0761i)14-s + (0.924 − 1.20i)15-s + (1.13 − 0.822i)16-s + (0.206 + 0.0326i)17-s + (−2.04 − 1.04i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.986 - 0.166i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162018 + 1.93804i\)
\(L(\frac12)\) \(\approx\) \(0.162018 + 1.93804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.02 + 0.958i)T \)
11 \( 1 \)
good2 \( 1 + (0.391 + 2.47i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (-2.33 + 1.19i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-0.167 + 0.328i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (4.37 - 0.693i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.849 - 0.134i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-1.37 + 4.22i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.17 + 2.17i)T + 23iT^{2} \)
29 \( 1 + (-0.237 - 0.731i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.66 - 2.66i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (9.76 + 4.97i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-4.76 - 1.54i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-4.69 + 4.69i)T - 43iT^{2} \)
47 \( 1 + (-4.90 - 9.63i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (0.157 + 0.993i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (1.04 - 0.339i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.67 - 9.18i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.17 + 4.17i)T - 67iT^{2} \)
71 \( 1 + (1.57 - 1.14i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.4 - 5.84i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-10.0 - 7.27i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.11 - 7.05i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 8.87iT - 89T^{2} \)
97 \( 1 + (7.09 - 1.12i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.08727987713017840442503511693, −9.288925900538314346181484714730, −8.925313360829420625519395224281, −7.946505629024707012711951300380, −6.90489870569121869476216386807, −5.17552736392137357347387119767, −4.05801701460586569969668519897, −2.69971541753557681190256009887, −2.30685524966714960723515898058, −1.10029388082280527418021665097, 2.32193811115914738703828839344, 3.67125249725756539646458629295, 4.93913739933643094107731464694, 5.72391477287077095309143802200, 6.81883811795273072938061956944, 7.70227882220543693927754163038, 8.336962314375680368143393148362, 9.288121693871346464189798515287, 9.772580927427616850839118448163, 10.36288892023036453805180585637

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