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Properties

Label 1-304-304.99-r1-0-0
Degree $1$
Conductor $304$
Sign $0.639 + 0.769i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.639 + 0.769i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.639 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230712494 + 0.5773857212i\)
\(L(\frac12)\) \(\approx\) \(1.230712494 + 0.5773857212i\)
\(L(1)\) \(\approx\) \(0.9125946177 + 0.1922564287i\)
\(L(1)\) \(\approx\) \(0.9125946177 + 0.1922564287i\)

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

−24.88247183931808373280726014350, −24.12147893167109932890914337663, −23.14060367633395677583989987244, −22.71676282709379944168933859871, −21.21144385182116356960859827472, −20.37215969291825424149057354424, −19.27988412561155450975499877003, −18.73458972438312635073361006404, −18.17069201325900108923772018845, −16.70079460729825218872491752040, −15.72218653081696239252511639213, −14.93174144722478846688920179491, −13.89933186130597271794348437515, −12.791663313245330698751165239514, −12.090077581194629480766599366753, −11.36131552924423719877422091629, −9.84240738920594887861044593096, −8.695515831585280526943975664425, −7.89367685344130983047482028986, −6.95607185329597378905605219445, −6.00123949636716482702140479477, −4.542047128157615261463032647702, −3.106235879349106847402184044361, −2.30473011924506212108236102766, −0.5899713280546325488214806692, 0.77491145686777200645333143913, 3.08432787410251256582103066670, 3.59437599651488302543529562874, 4.74217243555590374935466122999, 5.83107055590296027702289746385, 7.521818555552507580025673519665, 8.12695067456017478414091537880, 9.24731704636777230980324975397, 10.616273442316971454023284936847, 10.74185816312704938331732755974, 12.294033966375078462161254280386, 13.30077460515183316030260525594, 14.31401426286995847838125153455, 15.41201036401631960945592110313, 16.005210083300148078067172846638, 16.69699797602638625889555923075, 17.9087366066408488994287771382, 19.45389163329473897201083772841, 19.65108105929589628436171903982, 20.797624885867251553146286318424, 21.40930552685865712998633254924, 22.820755227628556244883160461640, 23.16109771493652779841555403224, 24.19273748313452705794193960640, 25.566799257684990306778891888

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