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Properties

Label 1-624-624.251-r0-0-0
Degree $1$
Conductor $624$
Sign $0.785 + 0.618i$
Analytic cond. $2.89784$
Root an. cond. $2.89784$
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  + i·5-s + (0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.5 + 0.866i)23-s − 25-s + (0.866 − 0.5i)29-s + 31-s + (0.866 + 0.5i)35-s + (−0.866 + 0.5i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + i·5-s + (0.5 − 0.866i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (0.5 + 0.866i)23-s − 25-s + (0.866 − 0.5i)29-s + 31-s + (0.866 + 0.5i)35-s + (−0.866 + 0.5i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.785 + 0.618i$
Analytic conductor: \(2.89784\)
Root analytic conductor: \(2.89784\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 624,\ (0:\ ),\ 0.785 + 0.618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.361908609 + 0.4720656116i\)
\(L(\frac12)\) \(\approx\) \(1.361908609 + 0.4720656116i\)
\(L(1)\) \(\approx\) \(1.116798313 + 0.1689232736i\)
\(L(1)\) \(\approx\) \(1.116798313 + 0.1689232736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + iT \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−22.96174241715011727021108045235, −21.88648750634079178120257468394, −21.11167093912169687735368907620, −20.70818406398153556231383212828, −19.53995431763226046284635189873, −18.82167730100840345936341964038, −17.85967345641013581403975780270, −17.16819519211709657707550175790, −16.03997946459627291464558790002, −15.65446631174155023618179657731, −14.518225469505877842072492086687, −13.61128050878290510931131321097, −12.62849500976327641253144256653, −12.11289147828724428520861971116, −11.05769141805085192714353740493, −10.10460545579606118202074574015, −8.90940575991467787593578835339, −8.45610070567745465254563083675, −7.54963717697016461174770579485, −6.09787885222305179435415754332, −5.291500168395258746444584417, −4.62806538073837119828149078445, −3.218155438022041932223568231999, −2.12120108940017548333997621, −0.87568155496482351099430849582, 1.1646214959645485596015303116, 2.558351703418256597485477489387, 3.402575754769575055345033920958, 4.59371713941824510335606351215, 5.524228023551525693788544695700, 6.79780782356987071776710658085, 7.48092947828693788809055832240, 8.15191744938308794887791103524, 9.82928704211076660780222477043, 10.14009386608915658424008885012, 11.23969524990097202714427885480, 11.84603908924156842079320536197, 13.2010335698998128953499079895, 13.92409877604965704572226064421, 14.60247106783712190596389138235, 15.55939942269725541000609319781, 16.3276129138138539816225083441, 17.62248621529293272307419379715, 17.91327103177840990823370454709, 18.936610316149074029068375101832, 19.71097232022956429129794559265, 20.866674397876199745306781584920, 21.139384361509941441004372385738, 22.54750604887214557617758267128, 22.9757031117652863818898992845

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