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Properties

Label 2-896-896.629-c0-0-0
Degree $2$
Conductor $896$
Sign $-0.0490 - 0.998i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 + 0.555i)2-s + (0.382 − 0.923i)4-s + (−0.195 + 0.980i)7-s + (0.195 + 0.980i)8-s + (0.980 − 0.195i)9-s + (−1.47 + 1.21i)11-s + (−0.382 − 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (0.555 − 1.83i)22-s + (1.63 + 1.08i)23-s + (−0.555 − 0.831i)25-s + (0.831 + 0.555i)28-s + (−0.598 + 0.728i)29-s + (0.980 + 0.195i)32-s + ⋯
L(s)  = 1  + (−0.831 + 0.555i)2-s + (0.382 − 0.923i)4-s + (−0.195 + 0.980i)7-s + (0.195 + 0.980i)8-s + (0.980 − 0.195i)9-s + (−1.47 + 1.21i)11-s + (−0.382 − 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (0.555 − 1.83i)22-s + (1.63 + 1.08i)23-s + (−0.555 − 0.831i)25-s + (0.831 + 0.555i)28-s + (−0.598 + 0.728i)29-s + (0.980 + 0.195i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.0490 - 0.998i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ -0.0490 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6285628752\)
\(L(\frac12)\) \(\approx\) \(0.6285628752\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 - 0.555i)T \)
7 \( 1 + (0.195 - 0.980i)T \)
good3 \( 1 + (-0.980 + 0.195i)T^{2} \)
5 \( 1 + (0.555 + 0.831i)T^{2} \)
11 \( 1 + (1.47 - 1.21i)T + (0.195 - 0.980i)T^{2} \)
13 \( 1 + (-0.555 + 0.831i)T^{2} \)
17 \( 1 + (0.707 - 0.707i)T^{2} \)
19 \( 1 + (-0.831 - 0.555i)T^{2} \)
23 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (0.598 - 0.728i)T + (-0.195 - 0.980i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.448 - 1.47i)T + (-0.831 + 0.555i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-1.26 - 0.124i)T + (0.980 + 0.195i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.124 - 0.151i)T + (-0.195 + 0.980i)T^{2} \)
59 \( 1 + (-0.555 - 0.831i)T^{2} \)
61 \( 1 + (0.980 - 0.195i)T^{2} \)
67 \( 1 + (0.195 + 1.98i)T + (-0.980 + 0.195i)T^{2} \)
71 \( 1 + (1.81 + 0.360i)T + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (-1.02 - 0.425i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (0.831 + 0.555i)T^{2} \)
89 \( 1 + (-0.382 + 0.923i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.25102503567878979693829451512, −9.615472202082594045307771738114, −8.985904027820063637053610013415, −7.84881854820558628488060420441, −7.34872140633227911458551831119, −6.41843715325356899072770525004, −5.35935432947737120782487027677, −4.69612016985515657193768295153, −2.83380656414832649389303683524, −1.72384267404992987022692851345, 0.848078503978862697565233712322, 2.46056788166653303947064380337, 3.52449190505501984167862585436, 4.51389693089646929955902272456, 5.85289831814389632365089038717, 7.18856637563298575250234467532, 7.52266187088505998418375621142, 8.477678377788241070143871138263, 9.385469708775904557075941750437, 10.26073484868003739769772789924

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