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Properties

Label 2-605-55.43-c1-0-0
Degree $2$
Conductor $605$
Sign $-0.389 - 0.921i$
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  + (−0.848 − 0.848i)2-s + (−0.258 − 0.258i)3-s − 0.560i·4-s + (−0.152 + 2.23i)5-s + 0.438i·6-s + (1.06 + 1.06i)7-s + (−2.17 + 2.17i)8-s − 2.86i·9-s + (2.02 − 1.76i)10-s + (−0.144 + 0.144i)12-s + (−3.20 + 3.20i)13-s − 1.80i·14-s + (0.616 − 0.537i)15-s + 2.56·16-s + (−2.50 − 2.50i)17-s + (−2.43 + 2.43i)18-s + ⋯
L(s)  = 1  + (−0.599 − 0.599i)2-s + (−0.149 − 0.149i)3-s − 0.280i·4-s + (−0.0680 + 0.997i)5-s + 0.179i·6-s + (0.402 + 0.402i)7-s + (−0.768 + 0.768i)8-s − 0.955i·9-s + (0.639 − 0.557i)10-s + (−0.0418 + 0.0418i)12-s + (−0.887 + 0.887i)13-s − 0.482i·14-s + (0.159 − 0.138i)15-s + 0.641·16-s + (−0.608 − 0.608i)17-s + (−0.573 + 0.573i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.389 - 0.921i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.389 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113950 + 0.171823i\)
\(L(\frac12)\) \(\approx\) \(0.113950 + 0.171823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.152 - 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (0.848 + 0.848i)T + 2iT^{2} \)
3 \( 1 + (0.258 + 0.258i)T + 3iT^{2} \)
7 \( 1 + (-1.06 - 1.06i)T + 7iT^{2} \)
13 \( 1 + (3.20 - 3.20i)T - 13iT^{2} \)
17 \( 1 + (2.50 + 2.50i)T + 17iT^{2} \)
19 \( 1 + 7.43T + 19T^{2} \)
23 \( 1 + (-1.84 - 1.84i)T + 23iT^{2} \)
29 \( 1 + 0.772T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 + (2.92 - 2.92i)T - 37iT^{2} \)
41 \( 1 - 6.91iT - 41T^{2} \)
43 \( 1 + (0.500 - 0.500i)T - 43iT^{2} \)
47 \( 1 + (-5.46 + 5.46i)T - 47iT^{2} \)
53 \( 1 + (-7.53 - 7.53i)T + 53iT^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 + 3.31iT - 61T^{2} \)
67 \( 1 + (3.34 - 3.34i)T - 67iT^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (6.73 - 6.73i)T - 73iT^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + (-2.80 + 2.80i)T - 83iT^{2} \)
89 \( 1 + 9.96iT - 89T^{2} \)
97 \( 1 + (8.11 - 8.11i)T - 97iT^{2} \)
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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

−10.89633864323944107627247821155, −10.14530076297281786391790975464, −9.234785152638032769633535883995, −8.688847351379070381436849340119, −7.24514470863267693034960579193, −6.55320254214513586681242771154, −5.59665920357963860755851862675, −4.25026102502679345332280548023, −2.77785867576208205508218630929, −1.85932118879633171934519659687, 0.13205862956875308645582926382, 2.16177997190015920070824778109, 3.94544571676586224565488460913, 4.82048650200563383177729991841, 5.83010600386686592496489450144, 7.10403643646661752033252727285, 7.87044450648317943828332608750, 8.496035836056861301436895950506, 9.208456670744054931960306095527, 10.38463719603674175731282263946

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