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Properties

Label 2-605-55.8-c1-0-31
Degree $2$
Conductor $605$
Sign $0.797 - 0.602i$
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  + (2.40 + 0.380i)2-s + (−0.271 + 0.532i)3-s + (3.72 + 1.21i)4-s + (1.85 − 1.25i)5-s + (−0.855 + 1.17i)6-s + (−2.30 + 1.17i)7-s + (4.15 + 2.11i)8-s + (1.55 + 2.13i)9-s + (4.92 − 2.31i)10-s + (−1.65 + 1.65i)12-s + (0.439 − 2.77i)13-s + (−5.97 + 1.94i)14-s + (0.166 + 1.32i)15-s + (2.84 + 2.06i)16-s + (−0.147 − 0.932i)17-s + (2.91 + 5.72i)18-s + ⋯
L(s)  = 1  + (1.69 + 0.269i)2-s + (−0.156 + 0.307i)3-s + (1.86 + 0.605i)4-s + (0.827 − 0.561i)5-s + (−0.349 + 0.480i)6-s + (−0.869 + 0.443i)7-s + (1.46 + 0.748i)8-s + (0.517 + 0.712i)9-s + (1.55 − 0.731i)10-s + (−0.478 + 0.478i)12-s + (0.121 − 0.769i)13-s + (−1.59 + 0.518i)14-s + (0.0429 + 0.342i)15-s + (0.710 + 0.516i)16-s + (−0.0358 − 0.226i)17-s + (0.687 + 1.34i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.797 - 0.602i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.797 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.66460 + 1.22876i\)
\(L(\frac12)\) \(\approx\) \(3.66460 + 1.22876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.85 + 1.25i)T \)
11 \( 1 \)
good2 \( 1 + (-2.40 - 0.380i)T + (1.90 + 0.618i)T^{2} \)
3 \( 1 + (0.271 - 0.532i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 + (2.30 - 1.17i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-0.439 + 2.77i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.147 + 0.932i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-1.28 - 3.94i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.104 - 0.104i)T + 23iT^{2} \)
29 \( 1 + (-2.14 + 6.60i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.33 - 5.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.44 + 6.77i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (3.27 - 1.06i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.91 + 3.91i)T - 43iT^{2} \)
47 \( 1 + (0.942 + 0.479i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (4.07 + 0.644i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-6.16 - 2.00i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.59 - 7.70i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.94 + 2.94i)T - 67iT^{2} \)
71 \( 1 + (1.02 + 0.744i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.804 + 1.57i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (3.51 - 2.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.18 + 1.29i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + 4.23iT - 89T^{2} \)
97 \( 1 + (2.25 - 14.2i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.81357368794015088804147393396, −10.05221854772372797969040151676, −9.125535208243851600265751773494, −7.80589483292549238638100931638, −6.74225223073797984680928784388, −5.71145614696626752648427478616, −5.40932734299953919755430256494, −4.32729773167678850495885356204, −3.25757138737883048994517327398, −2.07460430591963258390442327269, 1.71598432601016459466057422387, 3.02779437140431161191908037466, 3.80471869935644594026949783076, 4.96648945799487506077762511262, 6.04972309651695401231172769826, 6.70025547520426924093653695533, 7.12774370821586276177156926029, 9.127102638530023060151047438415, 9.912712057809680890374763012770, 10.87130234625031919609264511239

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