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Properties

Label 1-2608-2608.1205-r0-0-0
Degree $1$
Conductor $2608$
Sign $-0.707 - 0.706i$
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  + (−0.998 + 0.0581i)3-s + (0.342 − 0.939i)5-s + (0.286 + 0.957i)7-s + (0.993 − 0.116i)9-s + (0.998 − 0.0581i)11-s + (−0.642 − 0.766i)13-s + (−0.286 + 0.957i)15-s + (0.766 + 0.642i)17-s + (−0.727 + 0.686i)19-s + (−0.342 − 0.939i)21-s + (0.5 − 0.866i)23-s + (−0.766 − 0.642i)25-s + (−0.984 + 0.173i)27-s + (−0.230 − 0.973i)29-s + (−0.939 − 0.342i)31-s + ⋯
L(s)  = 1  + (−0.998 + 0.0581i)3-s + (0.342 − 0.939i)5-s + (0.286 + 0.957i)7-s + (0.993 − 0.116i)9-s + (0.998 − 0.0581i)11-s + (−0.642 − 0.766i)13-s + (−0.286 + 0.957i)15-s + (0.766 + 0.642i)17-s + (−0.727 + 0.686i)19-s + (−0.342 − 0.939i)21-s + (0.5 − 0.866i)23-s + (−0.766 − 0.642i)25-s + (−0.984 + 0.173i)27-s + (−0.230 − 0.973i)29-s + (−0.939 − 0.342i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2608\)    =    \(2^{4} \cdot 163\)
Sign: $-0.707 - 0.706i$
Analytic conductor: \(12.1115\)
Root analytic conductor: \(12.1115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2608} (1205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2608,\ (0:\ ),\ -0.707 - 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2527167501 - 0.6107805343i\)
\(L(\frac12)\) \(\approx\) \(0.2527167501 - 0.6107805343i\)
\(L(1)\) \(\approx\) \(0.7647594736 - 0.1406232215i\)
\(L(1)\) \(\approx\) \(0.7647594736 - 0.1406232215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
163 \( 1 \)
good3 \( 1 + (-0.998 + 0.0581i)T \)
5 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (0.286 + 0.957i)T \)
11 \( 1 + (0.998 - 0.0581i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.727 + 0.686i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-0.984 + 0.173i)T \)
41 \( 1 + (-0.893 + 0.448i)T \)
43 \( 1 + (-0.549 - 0.835i)T \)
47 \( 1 + (0.973 + 0.230i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + iT \)
61 \( 1 + (-0.642 - 0.766i)T \)
67 \( 1 + (-0.448 - 0.893i)T \)
71 \( 1 + (0.835 - 0.549i)T \)
73 \( 1 + (-0.893 - 0.448i)T \)
79 \( 1 + (-0.993 - 0.116i)T \)
83 \( 1 + (-0.957 + 0.286i)T \)
89 \( 1 + (0.0581 - 0.998i)T \)
97 \( 1 + (-0.993 - 0.116i)T \)
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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

−19.42222393820633332751785420317, −18.89235564823623353108310583341, −18.10249859160962322344409447060, −17.29587712395752603381764786683, −17.070552721146207059510201861048, −16.327758088799657742891600838793, −15.356915803883815494468805971536, −14.48743032881512950709534760341, −14.07580078633391461410508091357, −13.231935271840340698716188412104, −12.33164213826224851363586576227, −11.509756272987470185088617070664, −11.11157644738310962555441228718, −10.34354885498978119829899331587, −9.71141217222340324448851403404, −8.931900904056758616997207114876, −7.41373494801490697866024702519, −7.08788222898682433153601739117, −6.61416552964431086774746266338, −5.57686237018703666280264948324, −4.83905031792154153023354269985, −3.98841131895486377197380811275, −3.20713131140268709426704472386, −1.86668945044240517087939221208, −1.24182881370289305942641117264, 0.24548078991317273432264998840, 1.458103390827228756485756590705, 2.01112836488864100878389733787, 3.4330286361402517465147407604, 4.41922632488017915005319892406, 5.051310998543704784166852506686, 5.85956486931982838062088779174, 6.16160813555095719224493645395, 7.33483580483042941504246581466, 8.28639091332138462890845900648, 8.90520085252210958075071712922, 9.78724878169774891542634997546, 10.35303910209686810601883915826, 11.32865185731258999976944840405, 12.17153641161907643864939894893, 12.39829321909731428916666893726, 13.047935757124298917402883596460, 14.18802103913507951681838123754, 15.02651962406490048496822261684, 15.50737472759589032552400717620, 16.649285167841807355470952453941, 16.943680869011928191743816660736, 17.41183815120880449190203535094, 18.38084884291139044321190783583, 18.93227772098489911923831750382

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