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Properties

Label 2-605-55.18-c1-0-11
Degree $2$
Conductor $605$
Sign $-0.295 + 0.955i$
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  + (−0.396 − 2.50i)2-s + (−1.78 + 0.910i)3-s + (−4.21 + 1.36i)4-s + (−2.22 − 0.188i)5-s + (2.98 + 4.11i)6-s + (−0.820 + 1.61i)7-s + (2.79 + 5.48i)8-s + (0.599 − 0.824i)9-s + (0.412 + 5.65i)10-s + (6.27 − 6.27i)12-s + (2.22 − 0.352i)13-s + (4.35 + 1.41i)14-s + (4.15 − 1.69i)15-s + (5.46 − 3.96i)16-s + (−3.32 − 0.526i)17-s + (−2.30 − 1.17i)18-s + ⋯
L(s)  = 1  + (−0.280 − 1.77i)2-s + (−1.03 + 0.525i)3-s + (−2.10 + 0.684i)4-s + (−0.996 − 0.0840i)5-s + (1.21 + 1.67i)6-s + (−0.310 + 0.608i)7-s + (0.987 + 1.93i)8-s + (0.199 − 0.274i)9-s + (0.130 + 1.78i)10-s + (1.81 − 1.81i)12-s + (0.617 − 0.0978i)13-s + (1.16 + 0.378i)14-s + (1.07 − 0.436i)15-s + (1.36 − 0.991i)16-s + (−0.805 − 0.127i)17-s + (−0.542 − 0.276i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.295 + 0.955i$
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: \((2,\ 605,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259380 - 0.351681i\)
\(L(\frac12)\) \(\approx\) \(0.259380 - 0.351681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 + 0.188i)T \)
11 \( 1 \)
good2 \( 1 + (0.396 + 2.50i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (1.78 - 0.910i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (0.820 - 1.61i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (-2.22 + 0.352i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (3.32 + 0.526i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-0.403 + 1.24i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.16 + 3.16i)T + 23iT^{2} \)
29 \( 1 + (-0.907 - 2.79i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.24 - 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.85 + 1.45i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-1.23 - 0.402i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-4.55 + 4.55i)T - 43iT^{2} \)
47 \( 1 + (3.50 + 6.88i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (0.996 + 6.29i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-11.7 + 3.82i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.47 - 7.53i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.46 - 2.46i)T - 67iT^{2} \)
71 \( 1 + (-5.47 + 3.98i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.41 - 4.28i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (0.926 + 0.673i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.63 - 10.3i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 3.85iT - 89T^{2} \)
97 \( 1 + (-0.768 + 0.121i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.67462782487810530689930855910, −9.930191617097024455690628309214, −8.828513137800604574499424185744, −8.331439042926140769382929356810, −6.70950411101400372272533824759, −5.36181827576196155830918704807, −4.45964491308343703292325590477, −3.63488535736278252428301546347, −2.42047294409759512150571697125, −0.54739464929350028702695230746, 0.70451566996236707057200669659, 3.80317100291892784013497232289, 4.73290005515148686647626910760, 5.90396675727214324195464815566, 6.48556522250535553186123758508, 7.21901710087465322080527014391, 7.933194732971590623396031094978, 8.760768236574378957096941496038, 9.852835606456519964658849180658, 10.97580804099753887355346195090

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