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Properties

Label 1-304-304.37-r1-0-0
Degree $1$
Conductor $304$
Sign $-0.382 - 0.923i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
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·3-s + i·5-s − 7-s − 9-s + i·11-s + i·13-s − 15-s + 17-s − i·21-s − 23-s − 25-s − i·27-s + i·29-s − 31-s − 33-s + ⋯
L(s)  = 1  + i·3-s + i·5-s − 7-s − 9-s + i·11-s + i·13-s − 15-s + 17-s − i·21-s − 23-s − 25-s − i·27-s + i·29-s − 31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3721904942 + 0.5570224384i\)
\(L(\frac12)\) \(\approx\) \(-0.3721904942 + 0.5570224384i\)
\(L(1)\) \(\approx\) \(0.6087459402 + 0.5279625547i\)
\(L(1)\) \(\approx\) \(0.6087459402 + 0.5279625547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + iT \)
11 \( 1 \)
13 \( 1 + iT \)
17 \( 1 \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 - T \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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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

−24.46933812766290097063287743637, −23.72712824967170213984765605461, −22.87878733143273813046884511762, −21.9078667182850619354645129103, −20.64596916079240745676828430686, −19.86873373239620340838346174822, −19.12141737123015549405561135609, −18.26734054488206409462751763041, −17.08810229483442089172390418949, −16.48591134213551408633477883050, −15.475537062110134143608327773983, −13.98837439435936739998100960559, −13.297735750780774121334793597378, −12.506552840145682663299314910836, −11.82965664438173356033046626145, −10.41053323231124701802918058020, −9.234059173015499553025494766607, −8.274898411256658300938745387270, −7.48381572042141776327815676669, −5.98761300213201756818046008170, −5.60927785058479560437618170909, −3.77765996640133288018443723594, −2.66212053316190117045815464903, −1.09563229424240194572281686724, −0.22044612980683978031034534615, 2.20561657513361706339040192950, 3.37614036202122890186528916797, 4.15507981538963553880220430384, 5.58217823319351227309304467479, 6.596956579524593715273578501264, 7.59741081997522158220558722013, 9.19852566551714713924012776248, 9.82816588008071938376960047766, 10.606460909281413511537157332794, 11.66565293528953855280665154392, 12.70351425279546606559179737547, 14.24478452448839610703335441324, 14.58849494921174604854480351169, 15.81883508143513592164985021535, 16.35077292933232620445748784612, 17.50018260115154458645931481679, 18.54574065887493475229431940900, 19.48824157781786888926527272794, 20.30880373762776314531760173074, 21.48604310533358146889821538120, 22.07666583663963038711233511455, 22.93003631876316393198428180601, 23.51343408611127582868934408976, 25.24923294536407944621581217279, 26.0069251858308316313242331994

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