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Properties

Label 2-605-55.32-c1-0-40
Degree $2$
Conductor $605$
Sign $-0.163 + 0.986i$
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  + (1.29 − 1.29i)2-s + (2.21 − 2.21i)3-s − 1.36i·4-s + (0.668 + 2.13i)5-s − 5.73i·6-s + (−0.351 + 0.351i)7-s + (0.824 + 0.824i)8-s − 6.77i·9-s + (3.63 + 1.90i)10-s + (−3.01 − 3.01i)12-s + (−1.87 − 1.87i)13-s + 0.912i·14-s + (6.19 + 3.23i)15-s + 4.86·16-s + (−1.50 + 1.50i)17-s + (−8.78 − 8.78i)18-s + ⋯
L(s)  = 1  + (0.917 − 0.917i)2-s + (1.27 − 1.27i)3-s − 0.682i·4-s + (0.298 + 0.954i)5-s − 2.34i·6-s + (−0.132 + 0.132i)7-s + (0.291 + 0.291i)8-s − 2.25i·9-s + (1.14 + 0.600i)10-s + (−0.870 − 0.870i)12-s + (−0.520 − 0.520i)13-s + 0.243i·14-s + (1.59 + 0.836i)15-s + 1.21·16-s + (−0.364 + 0.364i)17-s + (−2.06 − 2.06i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.163 + 0.986i$
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: \((2,\ 605,\ (\ :1/2),\ -0.163 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29138 - 2.70209i\)
\(L(\frac12)\) \(\approx\) \(2.29138 - 2.70209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.668 - 2.13i)T \)
11 \( 1 \)
good2 \( 1 + (-1.29 + 1.29i)T - 2iT^{2} \)
3 \( 1 + (-2.21 + 2.21i)T - 3iT^{2} \)
7 \( 1 + (0.351 - 0.351i)T - 7iT^{2} \)
13 \( 1 + (1.87 + 1.87i)T + 13iT^{2} \)
17 \( 1 + (1.50 - 1.50i)T - 17iT^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + (0.308 - 0.308i)T - 23iT^{2} \)
29 \( 1 - 1.55T + 29T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 + (1.73 + 1.73i)T + 37iT^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (2.43 + 2.43i)T + 43iT^{2} \)
47 \( 1 + (-1.38 - 1.38i)T + 47iT^{2} \)
53 \( 1 + (2.14 - 2.14i)T - 53iT^{2} \)
59 \( 1 + 0.104iT - 59T^{2} \)
61 \( 1 + 7.73iT - 61T^{2} \)
67 \( 1 + (4.04 + 4.04i)T + 67iT^{2} \)
71 \( 1 + 7.94T + 71T^{2} \)
73 \( 1 + (-6.90 - 6.90i)T + 73iT^{2} \)
79 \( 1 + 9.59T + 79T^{2} \)
83 \( 1 + (-6.47 - 6.47i)T + 83iT^{2} \)
89 \( 1 - 8.24iT - 89T^{2} \)
97 \( 1 + (2.74 + 2.74i)T + 97iT^{2} \)
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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

−10.57768899247610922699248578136, −9.689340619700216907258156910916, −8.450611397113124771737045353726, −7.81127976925517721334737557088, −6.79752865392402947020798994113, −6.02266625691730240661038498229, −4.35867404444154223241235759147, −3.16833183970675454909959331026, −2.61466105219336173115329313708, −1.74700223477618493937456955763, 2.24288836652376824276791947155, 3.71299527791021218117936856999, 4.53355224896514816110667569079, 4.95291579543336258165553205937, 6.19415104666128245215131830839, 7.36269523598658211924878997257, 8.439348969483382577217683773387, 8.943827687618910154977577640850, 9.954342376202914615661890367371, 10.44291088247341195438204570207

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