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Properties

Label 1-304-304.299-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.0976 - 0.995i$
Analytic cond. $1.41177$
Root an. cond. $1.41177$
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  + (−0.342 + 0.939i)3-s + (−0.984 + 0.173i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (−0.342 − 0.939i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.642 − 0.766i)21-s + (0.173 − 0.984i)23-s + (0.939 − 0.342i)25-s + (0.866 − 0.5i)27-s + (−0.642 + 0.766i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (−0.984 + 0.173i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (−0.342 − 0.939i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.642 − 0.766i)21-s + (0.173 − 0.984i)23-s + (0.939 − 0.342i)25-s + (0.866 − 0.5i)27-s + (−0.642 + 0.766i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0976 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0976 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.0976 - 0.995i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.0976 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09207824576 - 0.1015502713i\)
\(L(\frac12)\) \(\approx\) \(0.09207824576 - 0.1015502713i\)
\(L(1)\) \(\approx\) \(0.5322664584 + 0.1620829706i\)
\(L(1)\) \(\approx\) \(0.5322664584 + 0.1620829706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−25.6641831909426760270182434881, −24.22167345173606287875529801021, −23.78992571637374106157817066445, −23.20624530309660206613816319520, −22.243143328078746073785574592439, −20.94538898885111152884026016534, −19.91637242435449008841173288369, −19.083066080077577445700985344387, −18.70446825660106552073703309959, −17.187716156195312573950530918049, −16.65836628671543726592347123359, −15.6914607109144564080071246787, −14.40751918276164109814130768874, −13.37633946208792811640993135243, −12.71258589995090886294341543325, −11.62432890541623021728183803785, −10.97077480603458034597454678537, −9.66841233029156116804363847415, −8.15685105920652902819690826763, −7.59063065343163715098485830695, −6.65898981962418651196034995998, −5.47253650959471801374105345805, −4.14610317295147394939341123930, −3.00691921985214992572387022914, −1.366751807121300959245263866373, 0.0966646774812105548995936634, 2.7309024682360577304298954273, 3.49290600819255762614138845192, 4.88455399573157039185800103193, 5.54924591323976789110657071806, 6.99866049114126914352059811553, 8.13675905613225719446692962976, 9.16764149321207191124069442438, 10.228633501105170703999894004633, 10.98027693811074728694336038780, 12.18927267531508658325824777708, 12.65337140967113986117055910203, 14.49400949893380210952077516889, 15.20397401144815703730460534874, 15.89679231550014193255531420654, 16.55844025346090563753035066708, 17.94789604567764702117944291900, 18.66947533669080840994995713289, 19.86173343189764075358257828722, 20.58455424024864116560558754139, 21.5700649131094375522677648262, 22.68775637031553186889067557077, 22.86108874213666275157513737320, 24.03488387255045789154003356324, 25.28549717394390894703670023599

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