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Properties

Label 2-605-55.18-c1-0-38
Degree $2$
Conductor $605$
Sign $0.496 + 0.867i$
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  + (1.45 − 0.743i)3-s + (1.90 − 0.618i)4-s + (0.459 − 2.18i)5-s + (−0.186 + 0.256i)9-s + (2.31 − 2.31i)12-s + (−0.956 − 3.53i)15-s + (3.23 − 2.35i)16-s + (−0.477 − 4.44i)20-s + (6.15 + 6.15i)23-s + (−4.57 − 2.01i)25-s + (−0.849 + 5.36i)27-s + (−8.04 − 5.84i)31-s + (−0.195 + 0.602i)36-s + (−10.6 − 5.44i)37-s + (0.474 + 0.525i)45-s + ⋯
L(s)  = 1  + (0.842 − 0.429i)3-s + (0.951 − 0.309i)4-s + (0.205 − 0.978i)5-s + (−0.0620 + 0.0853i)9-s + (0.668 − 0.668i)12-s + (−0.246 − 0.912i)15-s + (0.809 − 0.587i)16-s + (−0.106 − 0.994i)20-s + (1.28 + 1.28i)23-s + (−0.915 − 0.402i)25-s + (−0.163 + 1.03i)27-s + (−1.44 − 1.05i)31-s + (−0.0326 + 0.100i)36-s + (−1.75 − 0.894i)37-s + (0.0707 + 0.0782i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.496 + 0.867i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.496 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07234 - 1.20190i\)
\(L(\frac12)\) \(\approx\) \(2.07234 - 1.20190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.459 + 2.18i)T \)
11 \( 1 \)
good2 \( 1 + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (-1.45 + 0.743i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-6.15 - 6.15i)T + 23iT^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (8.04 + 5.84i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (10.6 + 5.44i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-1.72 - 3.38i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-2.13 - 13.4i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-3.15 + 1.02i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.52 + 1.52i)T - 67iT^{2} \)
71 \( 1 + (-2.42 + 1.76i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (-18.8 + 2.98i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.58771257484869393591902754151, −9.348755557270353757508856883752, −8.874915584100137807271932831065, −7.68546299479466915304434213492, −7.26035474538090735111221238762, −5.87143653292232838129732566234, −5.16219487388525313197641941329, −3.58214233928509648477638041953, −2.35825524287071893766424140068, −1.39595205095761063308323838418, 2.09604258842891188429395711304, 3.06216319122405785300711310463, 3.69665020543852768022957068706, 5.36100759582891600691951458007, 6.66939727137866937215236876592, 7.03488684847200171294585077770, 8.257795453950215746025799607326, 8.953075038691995364354492124912, 10.11983045564927281012843193820, 10.66581421374561681604427901471

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