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Properties

Label 2-605-11.9-c1-0-1
Degree $2$
Conductor $605$
Sign $-0.0219 + 0.999i$
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  + (−0.697 + 2.14i)2-s + (−0.628 + 0.456i)3-s + (−2.50 − 1.82i)4-s + (0.309 + 0.951i)5-s + (−0.542 − 1.66i)6-s + (0.100 + 0.0728i)7-s + (2.00 − 1.45i)8-s + (−0.740 + 2.27i)9-s − 2.25·10-s + 2.40·12-s + (−1.69 + 5.22i)13-s + (−0.226 + 0.164i)14-s + (−0.628 − 0.456i)15-s + (−0.184 − 0.566i)16-s + (−0.160 − 0.494i)17-s + (−4.37 − 3.17i)18-s + ⋯
L(s)  = 1  + (−0.493 + 1.51i)2-s + (−0.363 + 0.263i)3-s + (−1.25 − 0.910i)4-s + (0.138 + 0.425i)5-s + (−0.221 − 0.681i)6-s + (0.0379 + 0.0275i)7-s + (0.709 − 0.515i)8-s + (−0.246 + 0.759i)9-s − 0.714·10-s + 0.695·12-s + (−0.470 + 1.44i)13-s + (−0.0605 + 0.0439i)14-s + (−0.162 − 0.117i)15-s + (−0.0460 − 0.141i)16-s + (−0.0389 − 0.119i)17-s + (−1.03 − 0.749i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0219 + 0.999i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0219 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312921 - 0.319878i\)
\(L(\frac12)\) \(\approx\) \(0.312921 - 0.319878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.697 - 2.14i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.628 - 0.456i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.100 - 0.0728i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.69 - 5.22i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.160 + 0.494i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.55 + 1.85i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.92T + 23T^{2} \)
29 \( 1 + (3.29 + 2.39i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.17 + 6.70i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.10 - 5.16i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.10 - 4.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.42T + 43T^{2} \)
47 \( 1 + (0.369 - 0.268i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.0109 + 0.0337i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.42 - 3.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.37 + 7.31i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + (-3.79 - 11.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.89 - 5.00i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.93 - 5.96i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.193 + 0.595i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (0.567 - 1.74i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.41447786734137302131643344728, −9.892659005587751934449274984702, −9.653746048842882175171866928859, −8.384248782502415877168325726840, −7.74158287980696421111870337957, −6.80958413159474462663641601507, −6.09840160104178630064919196121, −5.17693727520008077535044905208, −4.25257255028070642416632261537, −2.30927211238331005644916849345, 0.30505539636165100213954874438, 1.55954161134380505660863728422, 2.95004584178363077502492813901, 3.87158509454793883976978380897, 5.28214616381723553287463092511, 6.22005606232602218514653197951, 7.65797487140808466878821018946, 8.520995511046442350845258687634, 9.440687039887216352670835843455, 10.10968735739485709168096610474

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