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Properties

Label 2-605-55.4-c1-0-28
Degree $2$
Conductor $605$
Sign $0.942 - 0.333i$
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  + (0.471 − 0.649i)2-s + (1.67 + 0.544i)3-s + (0.418 + 1.28i)4-s + (2.07 + 0.828i)5-s + (1.14 − 0.830i)6-s + (−0.563 + 0.182i)7-s + (2.56 + 0.832i)8-s + (0.0810 + 0.0589i)9-s + (1.51 − 0.958i)10-s + 2.38i·12-s + (1.05 − 1.45i)13-s + (−0.146 + 0.452i)14-s + (3.02 + 2.51i)15-s + (−0.444 + 0.322i)16-s + (−4.15 − 5.72i)17-s + (0.0765 − 0.0248i)18-s + ⋯
L(s)  = 1  + (0.333 − 0.459i)2-s + (0.966 + 0.314i)3-s + (0.209 + 0.644i)4-s + (0.928 + 0.370i)5-s + (0.466 − 0.339i)6-s + (−0.212 + 0.0691i)7-s + (0.905 + 0.294i)8-s + (0.0270 + 0.0196i)9-s + (0.479 − 0.302i)10-s + 0.689i·12-s + (0.292 − 0.402i)13-s + (−0.0392 + 0.120i)14-s + (0.781 + 0.649i)15-s + (−0.111 + 0.0807i)16-s + (−1.00 − 1.38i)17-s + (0.0180 − 0.00585i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.76084 + 0.474490i\)
\(L(\frac12)\) \(\approx\) \(2.76084 + 0.474490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.07 - 0.828i)T \)
11 \( 1 \)
good2 \( 1 + (-0.471 + 0.649i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.67 - 0.544i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.563 - 0.182i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.05 + 1.45i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.15 + 5.72i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.706 - 2.17i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.49iT - 23T^{2} \)
29 \( 1 + (-1.10 - 3.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.98 + 3.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.97 + 2.26i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.59 - 7.99i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.51iT - 43T^{2} \)
47 \( 1 + (1.83 + 0.596i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.40 + 1.92i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0118 - 0.0364i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.78 - 2.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.79iT - 67T^{2} \)
71 \( 1 + (-9.54 + 6.93i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.48 + 2.10i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.66 + 2.66i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.49 - 4.80i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (-3.15 + 4.34i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.78810647128339494648426121880, −9.688125848480032634296073319965, −9.117848548958333846968163982232, −8.172515631387949862844954878900, −7.22742088585139539199131482967, −6.20644765390924738092690908843, −4.93627609936710997643248364808, −3.67746181202141804746706154612, −2.89264688457398831401794389867, −2.08986550661704504597520316832, 1.58690563950973205226808075345, 2.47102740913968140116668320398, 4.09158270961608498062253698900, 5.19315518459588033200418422197, 6.21759348378369469266707999204, 6.77518605868371286235768782220, 8.033869239173789727663238311161, 8.842335554841344612200566446581, 9.571994405615412975725888381567, 10.52187749419255800792956398340

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