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Properties

Label 1-304-304.101-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.396 - 0.917i$
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  + (0.642 − 0.766i)3-s + (−0.342 − 0.939i)5-s + (0.5 + 0.866i)7-s + (−0.173 − 0.984i)9-s + (−0.866 − 0.5i)11-s + (−0.642 − 0.766i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (0.984 + 0.173i)21-s + (0.939 + 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)27-s + (0.984 − 0.173i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.342 − 0.939i)5-s + (0.5 + 0.866i)7-s + (−0.173 − 0.984i)9-s + (−0.866 − 0.5i)11-s + (−0.642 − 0.766i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (0.984 + 0.173i)21-s + (0.939 + 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)27-s + (0.984 − 0.173i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.396 - 0.917i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.396 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7150962400 - 1.087971876i\)
\(L(\frac12)\) \(\approx\) \(0.7150962400 - 1.087971876i\)
\(L(1)\) \(\approx\) \(1.014039595 - 0.5634514436i\)
\(L(1)\) \(\approx\) \(1.014039595 - 0.5634514436i\)

Euler product

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

−25.92877477065765753998045764173, −24.8653222722784582409350441960, −23.543657544468287432158898556817, −23.0745619333597010580392136165, −21.719677740588287572983847714153, −21.30265806779217533970696071218, −20.125454121982590767953445440318, −19.49634663613728695592248689069, −18.49686809612284357757956628675, −17.37259685126813671697311248548, −16.4109100423080646029715782487, −15.355088063957990081129955405786, −14.611777207273044393614016739908, −14.032480930615436727219422599560, −12.80962381268859407348753935005, −11.35312822014362878440602143231, −10.51771585345274637679870034918, −9.97925149107453154783981625439, −8.56259283098050527289341692441, −7.62390573724832096548874724862, −6.82159073790600250531384467730, −5.056548850228798962560313192028, −4.18574502418934895303325170416, −3.14664355670928615454807210304, −2.01240864162245381319063438367, 0.78583620116830319404308630243, 2.254518458842850299604059818354, 3.21002573298372385134035540359, 4.90464200831899090850654456522, 5.64557056203493834904032856948, 7.23805231866550922308029084337, 8.12005206895614622180040662385, 8.72604623971849540923166353870, 9.77806255712349520139731396912, 11.41211697350667766360076961172, 12.21144220325874814842670797572, 12.99661203390427785829443256564, 13.8365898840988406661816931440, 15.08922389439096786434863340114, 15.6635346689738979626170493617, 16.9669974490179892584794338263, 17.969027958454319866092770734836, 18.76141092126777874067625683736, 19.591990172751472572538077343304, 20.63869359951203543261096783905, 21.05597084170522209109260424428, 22.40989871530339182101458396803, 23.65265058791389900674462259691, 24.19037302918424910119578460489, 25.02423597540998399455108507662

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