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Properties

Label 2-61-61.3-c1-0-3
Degree $2$
Conductor $61$
Sign $0.247 + 0.968i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
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  + (−0.738 − 0.239i)2-s + (0.554 − 1.70i)3-s + (−1.13 − 0.821i)4-s + (−0.515 − 0.374i)5-s + (−0.819 + 1.12i)6-s + (1.31 − 0.427i)7-s + (1.55 + 2.13i)8-s + (−0.182 − 0.132i)9-s + (0.291 + 0.400i)10-s + 0.172i·11-s + (−2.02 + 1.47i)12-s + 2.97·13-s − 1.07·14-s + (−0.926 + 0.673i)15-s + (0.230 + 0.709i)16-s + (−4.72 + 6.49i)17-s + ⋯
L(s)  = 1  + (−0.522 − 0.169i)2-s + (0.320 − 0.986i)3-s + (−0.565 − 0.410i)4-s + (−0.230 − 0.167i)5-s + (−0.334 + 0.460i)6-s + (0.496 − 0.161i)7-s + (0.548 + 0.754i)8-s + (−0.0608 − 0.0442i)9-s + (0.0920 + 0.126i)10-s + 0.0519i·11-s + (−0.585 + 0.425i)12-s + 0.823·13-s − 0.286·14-s + (−0.239 + 0.173i)15-s + (0.0575 + 0.177i)16-s + (−1.14 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550044 - 0.427360i\)
\(L(\frac12)\) \(\approx\) \(0.550044 - 0.427360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-4.77 + 6.18i)T \)
good2 \( 1 + (0.738 + 0.239i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.554 + 1.70i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.515 + 0.374i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.31 + 0.427i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 - 0.172iT - 11T^{2} \)
13 \( 1 - 2.97T + 13T^{2} \)
17 \( 1 + (4.72 - 6.49i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.512 + 1.57i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.377 - 0.519i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 - 0.820iT - 29T^{2} \)
31 \( 1 + (-4.24 + 1.38i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (10.3 - 3.37i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.807 - 2.48i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-5.75 - 7.92i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + (2.67 + 3.68i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-7.74 - 2.51i)T + (47.7 + 34.6i)T^{2} \)
67 \( 1 + (2.71 - 3.73i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.82 + 5.27i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (10.0 - 7.33i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.32 + 1.82i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.38 + 4.25i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-10.1 - 3.30i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.79 - 11.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.60937941485123792326624081017, −13.58494005459001426086688980091, −12.86217149412017077545474366683, −11.32314549106791045277921239242, −10.23516504710467872662461889692, −8.622780394513453226865826424257, −8.043763088788298373742383209988, −6.39618123327196866863405805403, −4.48897561443367521807389403789, −1.63227788354796276569901765105, 3.62102493617919929483697705711, 4.86965510182611232326668398470, 7.14311208819630759031543324777, 8.566330196773182222358053537006, 9.294995726875220040873308279614, 10.47520922082344826139813721733, 11.73555737239280761377642363165, 13.28614904893062630458345634958, 14.28967113703463340254337568808, 15.65816513407065107747804791521

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