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Properties

Label 1-2608-2608.1323-r0-0-0
Degree $1$
Conductor $2608$
Sign $0.719 + 0.694i$
Analytic cond. $12.1115$
Root an. cond. $12.1115$
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  + (0.413 + 0.910i)3-s + (0.448 − 0.893i)5-s + (−0.466 + 0.884i)7-s + (−0.657 + 0.753i)9-s + (0.995 + 0.0968i)11-s + (0.802 + 0.597i)13-s + (0.999 + 0.0387i)15-s + (0.993 − 0.116i)17-s + (−0.305 − 0.952i)19-s + (−0.998 − 0.0581i)21-s + (0.939 − 0.342i)23-s + (−0.597 − 0.802i)25-s + (−0.957 − 0.286i)27-s + (−0.378 + 0.925i)29-s + (−0.0581 − 0.998i)31-s + ⋯
L(s)  = 1  + (0.413 + 0.910i)3-s + (0.448 − 0.893i)5-s + (−0.466 + 0.884i)7-s + (−0.657 + 0.753i)9-s + (0.995 + 0.0968i)11-s + (0.802 + 0.597i)13-s + (0.999 + 0.0387i)15-s + (0.993 − 0.116i)17-s + (−0.305 − 0.952i)19-s + (−0.998 − 0.0581i)21-s + (0.939 − 0.342i)23-s + (−0.597 − 0.802i)25-s + (−0.957 − 0.286i)27-s + (−0.378 + 0.925i)29-s + (−0.0581 − 0.998i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2608\)    =    \(2^{4} \cdot 163\)
Sign: $0.719 + 0.694i$
Analytic conductor: \(12.1115\)
Root analytic conductor: \(12.1115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2608} (1323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2608,\ (0:\ ),\ 0.719 + 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.195206874 + 0.8865175361i\)
\(L(\frac12)\) \(\approx\) \(2.195206874 + 0.8865175361i\)
\(L(1)\) \(\approx\) \(1.383349351 + 0.3536973535i\)
\(L(1)\) \(\approx\) \(1.383349351 + 0.3536973535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
163 \( 1 \)
good3 \( 1 + (0.413 + 0.910i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (-0.466 + 0.884i)T \)
11 \( 1 + (0.995 + 0.0968i)T \)
13 \( 1 + (0.802 + 0.597i)T \)
17 \( 1 + (0.993 - 0.116i)T \)
19 \( 1 + (-0.305 - 0.952i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.378 + 0.925i)T \)
31 \( 1 + (-0.0581 - 0.998i)T \)
37 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + (0.963 - 0.268i)T \)
43 \( 1 + (0.902 + 0.431i)T \)
47 \( 1 + (0.790 + 0.612i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.918 - 0.396i)T \)
67 \( 1 + (-0.968 - 0.249i)T \)
71 \( 1 + (0.996 - 0.0774i)T \)
73 \( 1 + (0.249 + 0.968i)T \)
79 \( 1 + (0.323 - 0.946i)T \)
83 \( 1 + (-0.845 + 0.533i)T \)
89 \( 1 + (0.0968 + 0.995i)T \)
97 \( 1 + (-0.981 + 0.192i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.176272619672209008717280930255, −18.7528001445083406079851846339, −17.89300484002224969585265076507, −17.2333153108231137594575456958, −16.729282181879775732994525113282, −15.618985573315011487343979083610, −14.65354590157102485242483278173, −14.30886717793913310369786512630, −13.57216297205991301275631618764, −13.04255013008340310254533547706, −12.19635235900854008098170982942, −11.37896214444468024534280286940, −10.586072768440388095020916978439, −9.90607587476533808436073211801, −9.11430016805955363306175420710, −8.18051536033691844716872946835, −7.45437064420146887451336001911, −6.80369579244228906924365003102, −6.18344591651984523216005428211, −5.55635169467397474018055124561, −3.864963214239863178880652479622, −3.49409479154726128794861176276, −2.68549283715211306422095527421, −1.52892983129832491740859853262, −0.94911584866701957612287448404, 0.95146498299601427711248403137, 2.041096879750839358031697217757, 2.860780453919231101746462462840, 3.839467443348916783593513000342, 4.45901458503878567755337194531, 5.41298655094551583430716317966, 5.90406962740400767335544390007, 6.877091735606559817591856813903, 8.06364282791066984425524633013, 9.005879329415398232891698829464, 9.124831800330516671138127947471, 9.682121196654621900454993111285, 10.8514159399276394793808659320, 11.43134676721882617579571695332, 12.43433607036035229755072689350, 12.93081073956005706401124056438, 13.93474546149717947962192394577, 14.44274726062665938951318287025, 15.27590347633913972394236107875, 15.99186855529618311205611339417, 16.520255850388969382402812973277, 17.09191890321470255818792981408, 17.93748510854662528497385674785, 19.0992821329184455741330691538, 19.321326574844086598465763058001

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