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Properties

Label 2-912-76.15-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.975 + 0.218i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (0.173 + 0.984i)3-s + (−0.345 + 0.290i)5-s + (−0.993 − 0.573i)7-s + (−0.939 + 0.342i)9-s + (1.09 − 0.630i)11-s + (−6.39 − 1.12i)13-s + (−0.345 − 0.290i)15-s + (−4.00 − 1.45i)17-s + (2.14 + 3.79i)19-s + (0.392 − 1.07i)21-s + (−5.76 + 6.87i)23-s + (−0.832 + 4.72i)25-s + (−0.5 − 0.866i)27-s + (−2.29 − 6.30i)29-s + (2.64 − 4.57i)31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (−0.154 + 0.129i)5-s + (−0.375 − 0.216i)7-s + (−0.313 + 0.114i)9-s + (0.329 − 0.189i)11-s + (−1.77 − 0.312i)13-s + (−0.0892 − 0.0749i)15-s + (−0.972 − 0.353i)17-s + (0.492 + 0.870i)19-s + (0.0856 − 0.235i)21-s + (−1.20 + 1.43i)23-s + (−0.166 + 0.944i)25-s + (−0.0962 − 0.166i)27-s + (−0.425 − 1.17i)29-s + (0.474 − 0.822i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.975 + 0.218i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.975 + 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0201438 - 0.182533i\)
\(L(\frac12)\) \(\approx\) \(0.0201438 - 0.182533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-2.14 - 3.79i)T \)
good5 \( 1 + (0.345 - 0.290i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.993 + 0.573i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.09 + 0.630i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.39 + 1.12i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (4.00 + 1.45i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.76 - 6.87i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (2.29 + 6.30i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.13iT - 37T^{2} \)
41 \( 1 + (3.40 - 0.601i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (6.46 + 7.70i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-3.37 - 9.26i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (7.68 - 9.15i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.0 - 3.65i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.87 + 4.93i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-6.07 + 2.21i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.08 - 5.10i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.463 - 2.63i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.29 + 13.0i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.21 + 1.85i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.48 - 1.67i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.447 - 1.22i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.25888815976489922495213889118, −9.754789047972235438333700679446, −9.148552177109645854796153697688, −7.83285331056182384154498827480, −7.37704849657544171439464887881, −6.14317602690351516166811983123, −5.26691479082399656039663076289, −4.21706100081045156438131127717, −3.35635083213912580814247381854, −2.13081046868130996776967504635, 0.07826891474763665629754150204, 1.98473891963376597166769600781, 2.90619450826806114417644622789, 4.36504501043118603445340665509, 5.13512069811323621469689545515, 6.61862541564331808126881840099, 6.80563014411467568785364404376, 8.014259495387342752474211380336, 8.732458434076493129780281351226, 9.637304093450179245279322009499

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