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Properties

Label 2-616-7.2-c1-0-19
Degree $2$
Conductor $616$
Sign $-0.947 - 0.318i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
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  + (−1.20 − 2.09i)3-s + (0.707 − 1.22i)5-s + (−1.62 − 2.09i)7-s + (−1.41 + 2.44i)9-s + (−0.5 − 0.866i)11-s − 3.82·13-s − 3.41·15-s + (1 + 1.73i)17-s + (2.70 − 4.68i)19-s + (−2.41 + 5.91i)21-s + (−1.29 + 2.23i)23-s + (1.50 + 2.59i)25-s − 0.414·27-s + 29-s + (−0.828 − 1.43i)31-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)3-s + (0.316 − 0.547i)5-s + (−0.612 − 0.790i)7-s + (−0.471 + 0.816i)9-s + (−0.150 − 0.261i)11-s − 1.06·13-s − 0.881·15-s + (0.242 + 0.420i)17-s + (0.621 − 1.07i)19-s + (−0.526 + 1.29i)21-s + (−0.269 + 0.466i)23-s + (0.300 + 0.519i)25-s − 0.0797·27-s + 0.185·29-s + (−0.148 − 0.257i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $-0.947 - 0.318i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ -0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105424 + 0.644618i\)
\(L(\frac12)\) \(\approx\) \(0.105424 + 0.644618i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.707 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.70 + 4.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.29 - 2.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (0.828 + 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.53 - 4.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-0.414 + 0.717i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.70 + 4.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.792 + 1.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.91 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.03 - 5.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + (-0.292 - 0.507i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.03 + 5.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.48T + 83T^{2} \)
89 \( 1 + (-7.41 + 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−10.10715947983601034431697871272, −9.406421339519016094105137607181, −8.170192198304330502897564143823, −7.22525180843555166633406375879, −6.72762034231786950890023965753, −5.65945098163931676913637486432, −4.80176709734628478224511175332, −3.22620695654665631689476086268, −1.65197379828808240891971839751, −0.38328396931174052839638251591, 2.43383566546140164621026429816, 3.54859808829563826622730859726, 4.80843712905335311978577114344, 5.52849839273620664665972224100, 6.39182043058938636872172565420, 7.47032624464068063795125073115, 8.774866027155355314252875693729, 9.874815623520930653740215039531, 9.977931363595993574810723417372, 10.88239299164855479398454746143

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