Properties

Label 2-605-55.7-c1-0-31
Degree $2$
Conductor $605$
Sign $0.789 - 0.614i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.50 − 0.396i)2-s + (0.910 + 1.78i)3-s + (4.21 − 1.36i)4-s + (−0.509 + 2.17i)5-s + (2.98 + 4.11i)6-s + (−1.61 − 0.820i)7-s + (5.48 − 2.79i)8-s + (−0.599 + 0.824i)9-s + (−0.412 + 5.65i)10-s + (6.27 + 6.27i)12-s + (−0.352 − 2.22i)13-s + (−4.35 − 1.41i)14-s + (−4.35 + 1.07i)15-s + (5.46 − 3.96i)16-s + (0.526 − 3.32i)17-s + (−1.17 + 2.30i)18-s + ⋯
L(s)  = 1  + (1.77 − 0.280i)2-s + (0.525 + 1.03i)3-s + (2.10 − 0.684i)4-s + (−0.227 + 0.973i)5-s + (1.21 + 1.67i)6-s + (−0.608 − 0.310i)7-s + (1.93 − 0.987i)8-s + (−0.199 + 0.274i)9-s + (−0.130 + 1.78i)10-s + (1.81 + 1.81i)12-s + (−0.0978 − 0.617i)13-s + (−1.16 − 0.378i)14-s + (−1.12 + 0.276i)15-s + (1.36 − 0.991i)16-s + (0.127 − 0.805i)17-s + (−0.276 + 0.542i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.789 - 0.614i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (282, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.789 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.08724 + 1.40350i\)
\(L(\frac12)\) \(\approx\) \(4.08724 + 1.40350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.509 - 2.17i)T \)
11 \( 1 \)
good2 \( 1 + (-2.50 + 0.396i)T + (1.90 - 0.618i)T^{2} \)
3 \( 1 + (-0.910 - 1.78i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (1.61 + 0.820i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (0.352 + 2.22i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.526 + 3.32i)T + (-16.1 - 5.25i)T^{2} \)
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 \( 1 + (-1.45 + 2.85i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-1.23 - 0.402i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (4.55 + 4.55i)T + 43iT^{2} \)
47 \( 1 + (-6.88 + 3.50i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (6.29 - 0.996i)T + (50.4 - 16.3i)T^{2} \)
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 \( 1 + (-4.28 + 8.41i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-0.926 - 0.673i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.3 - 1.63i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 - 3.85iT - 89T^{2} \)
97 \( 1 + (-0.121 - 0.768i)T + (-92.2 + 29.9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.72474906502933203816790897126, −10.24366291639294940259456534863, −9.398001940024634704803678829333, −7.80176926487027416429282902160, −6.84296626224439864783329500801, −5.98478949049281703005184283164, −4.91324913029649291159358416344, −3.85820471266634196256850108449, −3.38692450285055822812297481901, −2.50884943768192289949043770204, 1.74478697295595390261424468046, 2.85624755710038237452217468730, 4.06640973550908042599239149877, 4.89770601828264763052560135023, 6.07830250536019434059812821746, 6.65448920772449448457133696634, 7.70819531815603647306133311958, 8.410964431876519878353056392268, 9.575430527205579032344536935629, 11.04641773984093906664200720718

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