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Properties

Label 2-608-152.107-c1-0-5
Degree $2$
Conductor $608$
Sign $-0.133 - 0.991i$
Analytic cond. $4.85490$
Root an. cond. $2.20338$
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  + (−2.72 + 1.57i)3-s + (3.44 − 5.97i)9-s + 5.44·11-s + (3 + 5.19i)17-s + (−4.17 − 1.25i)19-s + (−2.5 + 4.33i)25-s + 12.2i·27-s + (−14.8 + 8.57i)33-s + (−9.39 + 5.42i)41-s + (5 + 8.66i)43-s + 7·49-s + (−16.3 − 9.43i)51-s + (13.3 − 3.14i)57-s + (1.62 − 0.937i)59-s + (14.1 + 8.18i)67-s + ⋯
L(s)  = 1  + (−1.57 + 0.908i)3-s + (1.14 − 1.99i)9-s + 1.64·11-s + (0.727 + 1.26i)17-s + (−0.957 − 0.287i)19-s + (−0.5 + 0.866i)25-s + 2.36i·27-s + (−2.58 + 1.49i)33-s + (−1.46 + 0.847i)41-s + (0.762 + 1.32i)43-s + 49-s + (−2.28 − 1.32i)51-s + (1.76 − 0.416i)57-s + (0.211 − 0.122i)59-s + (1.73 + 0.999i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ -0.133 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544965 + 0.623099i\)
\(L(\frac12)\) \(\approx\) \(0.544965 + 0.623099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (4.17 + 1.25i)T \)
good3 \( 1 + (2.72 - 1.57i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 5.44T + 11T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (9.39 - 5.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.62 + 0.937i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-14.1 - 8.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.84 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (4.89 + 2.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.151 + 0.0874i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.00576721023895121670435591368, −10.07797490250484249116722956184, −9.475612553673807927511865207052, −8.432631794125554361991011446940, −6.89813545384620417060057298945, −6.23262374339515358000903479567, −5.47253926000587687440145622160, −4.31296898406036545555056301892, −3.71323027671997443879947811694, −1.29666015847650864348056949022, 0.65623590961561566540050909059, 1.96033642778745147714369642078, 3.95501132887482202714171468417, 5.08788040296315106243305013678, 6.00247256004878400464877527703, 6.72375893593260465409279508386, 7.37163017782949387484662894738, 8.574605512919933104271754372019, 9.734858632469347358488503539421, 10.63469089304340636806596833880

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