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Properties

Label 2-605-55.32-c1-0-45
Degree $2$
Conductor $605$
Sign $-0.955 + 0.294i$
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  + (1.76 − 1.76i)2-s + (1.85 − 1.85i)3-s − 4.25i·4-s + (−2.19 + 0.412i)5-s − 6.56i·6-s + (−0.260 + 0.260i)7-s + (−3.98 − 3.98i)8-s − 3.88i·9-s + (−3.15 + 4.61i)10-s + (−7.89 − 7.89i)12-s + (3.13 + 3.13i)13-s + 0.922i·14-s + (−3.31 + 4.84i)15-s − 5.59·16-s + (−0.608 + 0.608i)17-s + (−6.87 − 6.87i)18-s + ⋯
L(s)  = 1  + (1.25 − 1.25i)2-s + (1.07 − 1.07i)3-s − 2.12i·4-s + (−0.982 + 0.184i)5-s − 2.67i·6-s + (−0.0985 + 0.0985i)7-s + (−1.41 − 1.41i)8-s − 1.29i·9-s + (−0.998 + 1.45i)10-s + (−2.27 − 2.27i)12-s + (0.868 + 0.868i)13-s + 0.246i·14-s + (−0.855 + 1.25i)15-s − 1.39·16-s + (−0.147 + 0.147i)17-s + (−1.62 − 1.62i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498318 - 3.31405i\)
\(L(\frac12)\) \(\approx\) \(0.498318 - 3.31405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.412i)T \)
11 \( 1 \)
good2 \( 1 + (-1.76 + 1.76i)T - 2iT^{2} \)
3 \( 1 + (-1.85 + 1.85i)T - 3iT^{2} \)
7 \( 1 + (0.260 - 0.260i)T - 7iT^{2} \)
13 \( 1 + (-3.13 - 3.13i)T + 13iT^{2} \)
17 \( 1 + (0.608 - 0.608i)T - 17iT^{2} \)
19 \( 1 - 4.44T + 19T^{2} \)
23 \( 1 + (2.17 - 2.17i)T - 23iT^{2} \)
29 \( 1 - 0.769T + 29T^{2} \)
31 \( 1 + 4.53T + 31T^{2} \)
37 \( 1 + (7.75 + 7.75i)T + 37iT^{2} \)
41 \( 1 + 5.01iT - 41T^{2} \)
43 \( 1 + (-4.69 - 4.69i)T + 43iT^{2} \)
47 \( 1 + (7.64 + 7.64i)T + 47iT^{2} \)
53 \( 1 + (-0.711 + 0.711i)T - 53iT^{2} \)
59 \( 1 + 1.09iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 + (-4.17 - 4.17i)T + 67iT^{2} \)
71 \( 1 - 1.94T + 71T^{2} \)
73 \( 1 + (-9.10 - 9.10i)T + 73iT^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (-5.04 - 5.04i)T + 83iT^{2} \)
89 \( 1 - 8.87iT - 89T^{2} \)
97 \( 1 + (-5.07 - 5.07i)T + 97iT^{2} \)
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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

−10.68994800551196268330244897032, −9.412758677997588246650313571019, −8.554949498520346895150773897315, −7.51155644885710287532437707362, −6.67416570170513533711505058831, −5.43088150346949114349796126922, −3.99807521701705008007593014949, −3.47537191094353586029741199017, −2.40840404680173878912394574862, −1.33015749043151101339799738124, 3.20962876275616019195232573328, 3.53319723536152442197437937198, 4.53287585771859140516333516107, 5.25533698186365943920689673723, 6.50916945622105096635481566534, 7.62141111112653679167814858292, 8.200147847259341692932020793217, 8.877696795051866880805299772599, 10.05136476753941318428748325421, 11.13952635772273480340452068497

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