Properties

Label 2-688-43.36-c1-0-13
Degree $2$
Conductor $688$
Sign $0.985 + 0.171i$
Analytic cond. $5.49370$
Root an. cond. $2.34386$
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  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (1.5 − 2.59i)7-s + (1 − 1.73i)9-s + (2.5 − 4.33i)13-s + (−0.499 + 0.866i)15-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + 3·21-s + (−3.5 − 6.06i)23-s + (2 − 3.46i)25-s + 5·27-s + (−1.5 + 2.59i)29-s + (2.5 + 4.33i)31-s + 3·35-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.566 − 0.981i)7-s + (0.333 − 0.577i)9-s + (0.693 − 1.20i)13-s + (−0.129 + 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + 0.654·21-s + (−0.729 − 1.26i)23-s + (0.400 − 0.692i)25-s + 0.962·27-s + (−0.278 + 0.482i)29-s + (0.449 + 0.777i)31-s + 0.507·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $0.985 + 0.171i$
Analytic conductor: \(5.49370\)
Root analytic conductor: \(2.34386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :1/2),\ 0.985 + 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88751 - 0.163164i\)
\(L(\frac12)\) \(\approx\) \(1.88751 - 0.163164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (-4 - 5.19i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.50394288321461305729714503499, −9.836442806056808110997380227951, −8.586794310775862755486423803224, −8.039889548999201075403027815518, −6.84630890041493936075302620159, −6.11242632792189946982289547879, −4.72839371823650277054352650689, −3.92527711155799805691515586164, −2.90669861799568570656320516641, −1.14482069443833873669511896856, 1.59879615059605037345712103737, 2.39507147899462068171661761233, 4.04726142858671480018569921069, 5.11700664022436826688220904845, 5.95506543807325213765945792119, 7.13130282015602544162713464820, 7.88222145262658277150844246147, 8.922763163572864049796116384908, 9.265678116799833646400514511573, 10.54367726674060662275796009401

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