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Properties

Label 2-656-41.21-c1-0-15
Degree $2$
Conductor $656$
Sign $-0.208 + 0.978i$
Analytic cond. $5.23818$
Root an. cond. $2.28870$
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  + (−0.0432 − 0.0432i)3-s + (−0.422 + 0.581i)5-s + (1.42 − 2.79i)7-s − 2.99i·9-s + (−3.06 − 0.486i)11-s + (−1.79 + 0.912i)13-s + (0.0434 − 0.00687i)15-s + (0.304 − 1.92i)17-s + (−3.91 − 1.99i)19-s + (−0.182 + 0.0592i)21-s + (−0.275 + 0.848i)23-s + (1.38 + 4.26i)25-s + (−0.259 + 0.259i)27-s + (−1.43 − 9.03i)29-s + (5.64 − 4.10i)31-s + ⋯
L(s)  = 1  + (−0.0249 − 0.0249i)3-s + (−0.188 + 0.259i)5-s + (0.537 − 1.05i)7-s − 0.998i·9-s + (−0.925 − 0.146i)11-s + (−0.496 + 0.253i)13-s + (0.0112 − 0.00177i)15-s + (0.0738 − 0.466i)17-s + (−0.897 − 0.457i)19-s + (−0.0398 + 0.0129i)21-s + (−0.0574 + 0.176i)23-s + (0.277 + 0.852i)25-s + (−0.0499 + 0.0499i)27-s + (−0.265 − 1.67i)29-s + (1.01 − 0.736i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.208 + 0.978i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ -0.208 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684532 - 0.845658i\)
\(L(\frac12)\) \(\approx\) \(0.684532 - 0.845658i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (2.53 + 5.88i)T \)
good3 \( 1 + (0.0432 + 0.0432i)T + 3iT^{2} \)
5 \( 1 + (0.422 - 0.581i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.42 + 2.79i)T + (-4.11 - 5.66i)T^{2} \)
11 \( 1 + (3.06 + 0.486i)T + (10.4 + 3.39i)T^{2} \)
13 \( 1 + (1.79 - 0.912i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.304 + 1.92i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (3.91 + 1.99i)T + (11.1 + 15.3i)T^{2} \)
23 \( 1 + (0.275 - 0.848i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.43 + 9.03i)T + (-27.5 + 8.96i)T^{2} \)
31 \( 1 + (-5.64 + 4.10i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.49 + 2.53i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-10.3 - 3.35i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (5.24 + 10.2i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.465 - 2.94i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (1.37 - 4.23i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.3 + 3.69i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.00 - 1.26i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-0.988 - 0.156i)T + (67.5 + 21.9i)T^{2} \)
73 \( 1 - 6.49iT - 73T^{2} \)
79 \( 1 + (-4.43 - 4.43i)T + 79iT^{2} \)
83 \( 1 + 7.83T + 83T^{2} \)
89 \( 1 + (6.40 - 12.5i)T + (-52.3 - 72.0i)T^{2} \)
97 \( 1 + (-6.34 + 1.00i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.31271115685149465121783331625, −9.560444780558295888902830218119, −8.480215275631061051922386077716, −7.52458842551497198726817388818, −6.94102812984901115750740150708, −5.79682446500792729097007028878, −4.60218137523885917458094132193, −3.75446040561950723281338340001, −2.41476400109087476735012674106, −0.57143600469239471041357993017, 1.90062278416239174208225939391, 2.89529852566108112898964602934, 4.63264472793715941745674652740, 5.15031266264007053090753649535, 6.17438350471167954538070613041, 7.48922651937848128498638030263, 8.300595256285496124927968137630, 8.748253810034291140693066791084, 10.16841758357963368988042510233, 10.63910576753539692118549355387

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