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Properties

Label 1-304-304.35-r1-0-0
Degree $1$
Conductor $304$
Sign $-0.917 - 0.396i$
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  + (−0.642 − 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

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

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.917 - 0.396i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ -0.917 - 0.396i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02219001756 + 0.1073014947i\)
\(L(\frac12)\) \(\approx\) \(0.02219001756 + 0.1073014947i\)
\(L(1)\) \(\approx\) \(0.6505363779 + 0.08304430238i\)
\(L(1)\) \(\approx\) \(0.6505363779 + 0.08304430238i\)

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

−24.54488449785104111229743149431, −23.48501302353989657391743444292, −22.80513939120300237610488458613, −22.09272224857885638147342332711, −20.859232760922955381951155348835, −20.15850375820730515540135042499, −19.5677030833770179523206132572, −17.872230706610720769237200291511, −17.15846286139226472255569706869, −16.398104139070821890854151196338, −15.737253390853012146568117062997, −14.62496929790807314252516988515, −13.46206600402767811011838868558, −12.222182300502677546168080697978, −11.80553499686734588496134220601, −10.30762331792897944226131518124, −9.78824055412520807948785902469, −8.695928622404065869803704275584, −7.37714090255507531286533230362, −6.294737636561172823065158758052, −4.96964425027012003566444967784, −4.3498377834736514301938287423, −3.23027043479162269924329392481, −1.07114453388113968327928686718, −0.041296146290769434272016694876, 1.74774535444846242241809682651, 2.86750956399035876035619290787, 4.22932591415554463391975621419, 5.89913081886037892608298748565, 6.37298147517188679024653374377, 7.39242708564010417774258119039, 8.496792270830708657528367520146, 9.78855541554501419964025832805, 10.9212040295253358938760872527, 11.88332423924134181037700797610, 12.31358310265444909463560669018, 13.704105788436482341454633962965, 14.51557244913336529281746732384, 15.60210941413997933518988110117, 16.61863854707220320041711636677, 17.53925213019882106988957156041, 18.51957008339767096722965194646, 19.22393081349508738329539588180, 19.65695162178898307477375649996, 21.75726652184053178933051542567, 21.92437742599996669442982334635, 22.90786221645990049864049707844, 23.80693902966187776207046688253, 24.611474861469608883575040762980, 25.52952632041631293151190665120

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